Introduction (5 pages): For a concise summary of the thesis, read the Introduction in English. It provides an overview of the research and its key points.

Overview (Chapter 1, 12 pages): If you require a formal summary of the main notions and results, refer to Chapter Overview. This section presents a comprehensive overview in a structured manner.

Preliminaries (Part I, 25 pages): To gain the necessary background knowledge to understand the results, delve into Part Preliminaries. This section provides the foundational concepts and context required for comprehension.

Résumé

L'

objectif principal de cette thèse est d'examiner le concept de "type calculable" et d'améliorer notre compréhension globale de cette notion, ainsi que de fournir des techniques pour vérifier ou réfuter cette propriété. Un espace métrisable compact est dit de type calculable si pour tout ensemble homéomorphe à cet espace, semi-calculabilité et calculabilité sont équivalentes. Cette étude s'appuie sur les travaux de Miller, qui a démontré que les sphères de dimension finie sont de type calculable, ainsi que sur les travaux d'Iljazović et d'autres auteurs, qui ont étendu cette propriété à divers espaces tels que les variétés compactes.

Pour commencer, nous établissons l'équivalence entre deux définitions distinctes de type calculable présentes dans la littérature, impliquant respectivement des espaces métriques et des espaces de Hausdorff. Nous soutenons que la version relativisée et plus forte du type calculable présente des propriétés plus favorables et se prête bien à l'analyse topologique. Nous obtenons ainsi des caractérisations du "type calculable fort" de nature purement topologiques, et en lien avec la complexité descriptive des invariants topologiques.

Cela nous amène naturellement à notre deuxième objectif, l'étude du pouvoir expressif des invariants topologiques de faible complexité descriptive et de leur capacité à distinguer des espaces différents. Plus précisément, nous étudions deux familles d'invariants topologiques de faible complexité qui capturent l'extensibilité et la trivialité homotopique des fonctions continues. En utilisant ce cadre, nous revisitons les résultats précédents sur le type calculable et découvrons de nouvelles perspectives. Notamment, nous identifions la complexité du problème de séparation des graphes topologiques finis.

Enfin, notre troisième objectif se concentre sur l'application de la théorie de l'homologie à l'étude de ce que nous appelons la "propriété de surjection", qui caractérise la propriété de type calculable. Par exemple, nous prouvons qu'un complexe simplicial fini est de type calculable (fort) si et seulement si l'étoile de chaque sommet satisfait la propriété de surjection. De plus, la réduction à l'homologie implique que la propriété de type calculable est décidable pour les complexes simpliciaux finis de dimension au plus 4.

i Abstract T he main objective of this thesis is to examine the concept of "computable type" and enhance our overall understanding of this notion, as well as provide techniques for verifying or disproving this property. A compact metrizable space is said to have computable type if any semicomputable homeomorphic copy of that space is actually computable. This study builds upon the work of Miller, who demonstrated that finite-dimensional spheres have computable type, and Iljazović and other authors, who extended this property to various spaces, including compact manifolds.

To begin, we establish the equivalence between two distinct definitions of computable type present in the literature, involving metric spaces and Hausdorff spaces, respectively. We contend that the stronger, relativized version of computable type exhibits more favorable properties and lends itself well to topological analysis. Consequently, we derive characterizations of "strong computable type" in purely topological terms as well as the descriptive complexity of topological invariants.

It naturally leads to our second objective, which is the study of the expressive power of topological invariants of low descriptive complexity and their ability to differentiate between spaces. Specifically, we investigate two families of low descriptive complexity topological invariants that capture the extensibility and null-homotopy of continuous functions. Using this framework, we revisit previous findings on computable type and discover new insights. Notably, we identify the complexity of the finite topological graph separation problem.

Lastly, our third objective revolves around applying homology theory to study what we term the "surjection property" which characterizes the computable type property. For instance, we prove that a finite simplicial complex has (strong) computable type if and only if the star at each vertex satisfies the surjection property. Furthermore, the reduction to homology implies that the computable type property is decidable, for finite simplicial complexes of dimension at most 4.

iii

Acknowledgment

A bove all, I would like to express my deepest gratitude to the Almighty God for establishing me, guiding me on the right path, inspiring me to take the right steps and the right reflexes, giving me the strength and the composure to complete this thesis and allowing me to fulfill my dreams... At the very beginning, I wish to express my sincere thanks to Mathieu Hoyrup for his exceptional supervision, unwavering guidance, invaluable support, constant availability, insightful remarks, and invaluable advice throughout the duration of these three years. It is thanks to his patient and comprehensive approach that I not only acquired a foundation in computability theory but also learned the art of becoming a researcher. It is worth noting that any mistakes made along the way are entirely my own.

I would like to express my sincere appreciation to my second supervisor, Emmanuel Jeandel, first for accepting my registration into the Ph.D. program. Throughout my academic journey, I have greatly valued the numerous insightful discussions I have had with him. His expertise, guidance, and thought-provoking conversations have played a vital role in shaping my research.

I would like to express my heartfelt thanks to Olivier Bournez and Vasco Brattka for serving as referees for my dissertation, and to Nathalie Aubrun, Laurent Bienvenu and Monique Teillaud for being part of my defense committee. Their contributions and expertise are invaluable for the completion and evaluation of my research. I thank Horatiu Cirstea and Monique Teillaud again for being members of my individual follow-up committee.

I wholeheartedly thank Pierre Guillon, without whom this thesis would not have been possible. I would like to thank Guilhem Gamard and Arno Pauly for engaging in discussions about some of our results. I thank Joseph Miller and Zvonko Iljazović for introducing the beautiful notion of computable type.

I take this opportunity to thank: Lionel Nguyen Van Thé, Pascal Vanier, Benoit Monin and Laurent Régnier for their help in my PhD application.

I would like to thank Lorraine University and LORIA laboratory for giving me the chance to do this PhD. Special thanks to our research team MOCQUA. I place on record, Michel Mehrenberger for accepting me to join the master degree and François Hamel for accepting me in Archimedes Institute scholarship and for their invaluable assistance; and Boban Velickovic for suggesting my master internship with Mathieu.

I also place on record, Tarik Ali-ziane and my master 1 professors Aissa Naima and Ouahiba Zair for their unvaluable help.

My special thanks go to my USTHB professors: Toufik Laadj, Ammar Khemmoudj, Mourad Rahmani, Mehdi Belraouti, Mohamed Belache, Salim Khelifa, Chahrazad Titri and Yazid Raffed for their help.

In addition to Abdelghani Zeghib, a source of motivation, the MathCamps leaders and the ones of African Institute for Mathematical Sciences, Nabila Belhamra for her peaces of advice, my high school teacher Djemmel, my college teacher Bouchina, my primary school teachers Redouani and Mustapha. Nabil Zahra and many of my professors during my studies.

I would like to wholeheartedly express my deepest appreciation, thanks, and utmost regards to my family. Their constant support, prayers, boundless love, unwavering tenderness, and invaluable advice have been instrumental in my entire life: v I would like to dedicate this achievement to my father, whose tireless sacrifices and selflessness over the years have paved the way for my progress in life. May God bless this endeavor and grant it success. I am deeply grateful for the noble values, education, strong moral compass and passion for knowledge and personal growth instilled in me by my father. His guidance and wisdom have been invaluable in guiding me towards a meaningful path. May God grant him eternal mercy and make the highest paradise his share. Thank you infinitely! I want to express my heartfelt appreciation and deepest gratitude to my mother, who dedicated herself to ensuring my success. I am grateful for the countless sacrifices she made, her endless support, and her constant presence in my life. From the smallest acts of care to the most significant challenges she has faced on my behalf, she has exemplified a love that knows no bounds. Through this work, humble as it may be, I hope to convey my profound emotions and eternal gratitude to her. May God protect you, prolong your life and increase your work. I love you so much!

Introduction in French

L'

analyse calculable fournit des notions de calculabilité pour divers objets mathématiques tels que les nombres réels, les ensembles, les fonctions, etc. Nous nous concentrons en particulier sur les sous-ensembles compacts des espaces euclidiens et des espaces topologiques plus généraux, pour lesquels les deux notions d'ensembles compacts calculables et ensembles compacts semi-calculables sont définies. En substance, un sous-ensemble du plan euclidien est considéré comme calculable s'il est possible de concevoir un programme capable de représenter l'ensemble sur un écran avec une résolution arbitraire. D'autre part, un sous-ensemble est considéré comme semi-calculable s'il existe un programme capable de rejeter les points se trouvant en-dehors de l'ensemble. Une illustration bien connue de ces concepts est l'ensemble de Mandelbrot, qui peut être facilement reconnu comme semi-calculable sur la base de sa définition. Cependant, déterminer sa calculabilité reste un problème ouvert, lié à une conjecture en dynamique complexe [START_REF] Hertling | Is the Mandelbrot set computable?[END_REF].

Les travaux pionniers de Miller [START_REF] Miller | Effectiveness for embedded spheres and balls[END_REF] ont révélé que, pour certains ensembles, la semi-calculabilité et la calculabilité sont, étonnamment, équivalentes. Il a prouvé que les sphères de dimension finie plongées dans les espaces euclidiens exhibent cette propriété remarquable. Par la suite, Iljazović a développé une une exploration systématique de cette propriété en se concentrant sur les sphères plongées dans les espaces métriques calculables [START_REF] Iljazović | spheres and cells in computable metric spaces[END_REF], sur les variétés compactes [START_REF] Iljazovic | Compact manifolds with computable boundaries[END_REF], et sur divers autres ensembles, comme en témoigne une série d'articles écrits en collaboration avec plusieurs chercheurs [START_REF] Iljazović | Chainable and circularly chainable co-r.e. sets in computable metric spaces[END_REF][START_REF] Iljazović | spheres and cells in computable metric spaces[END_REF][START_REF] Iljazovic | Compact manifolds with computable boundaries[END_REF][START_REF] Burnik | Computability of 1-manifolds[END_REF][START_REF] Iljazović | Semicomputable manifolds in computable topological spaces[END_REF][START_REF] Čičković | Chainable and circularly chainable semicomputable sets in computable topological spaces[END_REF][START_REF] Horvat | Computability of pseudo-cubes[END_REF][START_REF] Iljazović | Computability of graphs[END_REF][START_REF] Čelar | Computability of products of chainable continua[END_REF][START_REF] Čelar | Computability of glued manifolds[END_REF].

Ces travaux ont conduit à la définition du type calculable : un espace métrisable compact X est de type calculable si toutes les copies semi-calculables de cet espace dans le cube de Hilbert sont calculables (le cube de Hilbert étant un produit cartésien infini dénombrable d'intervalles fermés [0, 1]). Une paire compacte (X, A compacts avec A ⊂ X) est dite de type calculable si, pour toute copie (Y, B) de cette paire, si Y et B sont tout les deux semi-calculables, alors Y est calculable. Ainsi, le segment [0, 1] n'est pas de type calculable, mais la paire ([0, 1], {0, 1}) est de type calculable.

Cette propriété constitue le thème central de cette thèse. Notre objectif est d'entreprendre une investigation théorique complète de la notion de type calculable, dans le but d'améliorer notre compréhension de ce concept et de fournir une boîte à outils pour prouver ou réfuter cette propriété. Nous avons remarqué que les techniques existantes dans la littérature sont généralement très spécifiques et pourraient bénéficier de l'intégration d'arguments plus unifiés, ce qui nous mène à explorer de nouvelles approches.

Ce projet interdisciplinaire englobe l'analyse calculable, la théorie descriptive des ensembles et la topologie générale et algébrique. L'étude de la propriété de calculabilité en question interagit de manière complexe avec la topologie, ce qui rend essentielle son incorporation dans notre recherche.

Dans cette introduction, nous donnerons un aperçu succinct et informel des résultats que nous avons obtenus dans diverses directions. Une élaboration plus détaillée sera présentée dans les parties suivantes.

La notion de type calculable. Principalement, deux notions de type calculable ont été introduites dans la littérature, qui considèrent des copies de l'espace plongées dans différentes classes d'espaces topologiques, à savoir les espaces métriques calculables et les espaces de Hausdorff calculables. Plusieurs résultats ont été établis sur les espaces métriques calculables d'abord [START_REF] Iljazović | Chainable and circularly chainable co-r.e. sets in computable metric spaces[END_REF][START_REF] Iljazovic | Compact manifolds with computable boundaries[END_REF], puis étendus aux espaces de Hausdorff calculables [START_REF] Čičković | Chainable and circularly chainable semicomputable sets in computable topological spaces[END_REF][START_REF] Iljazović | Semicomputable manifolds in computable topological spaces[END_REF]. Nous montrons d'abord que ces deux notions sont en réalité équivalentes, et qu'il suffit de considérer des copies dans le cube de Hilbert (voir le théorème 4.2.1).

xv La notion de type calculable fort. Il n'est pas possible d'obtenir des caractérisations générales complètes de la propriété de type calculable, car la plupart des espaces sont trivialement de type calculable en raison de l'absence de copies semi-calculables. Pour surmonter cette limitation, nous introduisons une notion plus souple que nous appelons type calculable fort, qui correspond à un type calculable relatif à un oracle quelconque (voir la définition 4.3.1).

Avec la notion de type calculable fort, nous sommes en mesure d'obtenir de nombreuses caractérisations et résultats topologiques (voir les théorèmes 4.3.1 et 4. 3.3). De plus, cela révèle des liens étroits avec d'autres notions qui motivent une exploration indépendante et un développement ultérieur (voir le corollaire 4.4.3). Dans les deux paragraphes qui suivent, nous donnons quelques détails sur ces résultats.

Type calculable fort et minimalité. Notre première caractérisation est qu'un espace est de type calculable fort si et seulement s'il satisfait une propriété de faible complexité descriptive, et qu'il est minimal au sens où aucun sous-espace propre ne satisfait cette propriété (théorèmes 4.3.1 et 4. 3.3). En particulier, tout espace minimal satisfaisant un invariant Σ 0 2 est de type calculable fort (théorème 4.3.2). Cela sert de motivation pour rechercher des invariants Σ 0 2 , ce qui permet de revisiter les résultats précédents et d'en obtenir de nouveaux. Cette découverte élargit considérablement notre compréhension et fournit un modèle commun pour unifier les preuves de la propriété de type calculable fort.

Nous identifions des classes d'invariants Σ 0 2 impliquant des fonctions continues vers des sphères et, plus généralement, des compacts Absolute Neighborhood Retracts (ANRs), voir la section 5.2.3. Lorsque Y est un ANR compact fixé et X un espace compact variable, nous montrons comment l'ensemble [X; Y ] des fonctions continues de X vers Y , modulo l'homotopie, peut être calculé à partir de X, sans supposer la calculabilité de Y (théorème 5.2.1). Un cas particulièrement intéressant est lorsque Y est la sphère n-dimensionnelle S n . Cette analyse induit immédiatement des invariants topologiques de faible complexité descriptive, qui sont très classiques en topologie. Nous définissons les invariants E n et H n qui capturent l'extensibilité et la trivialité homotopique des fonctions continues vers la sphère n-dimensionnelle S n (voir les définitions 5.3.2 et 5. 3.3).

Nous étudions en détail ces invariants topologiques et revisitions plusieurs résultats de la littérature sur le type calculable en identifiant un invariant Σ 0 2 approprié pour lequel l'espace ou la paire est minimale (voir la section 5.4). Ces résultats montrent que la théorie s'applique effectivement et renforcent les résultats précédents car ils fournissent plus d'informations et peuvent être utilisés pour dériver de nouveaux résultats. Par exemple, La paire (B n+1 , S n ) est minimale par rapport à E n (proposition 5.3.6).

Toute variété fermée de dimension n est minimale par rapport à H n (théorème 5.4.1). La paire constituée de la "soucoupe de Varsovie" et de son bord circulaire est minimale par rapport à E 1 et est donc de type calculable fort (proposition 5.4.3), voir figure 1.

Notre cadre de travail permet souvent d'obtenir des preuves plus simples des résultats en divisant l'argument en deux parties : une composante liée à la théorie de la calculabilité (démontrant qu'un invariant topologique est Σ 0

2 ) et une composante purement topologique (établissant qu'un espace est minimal par rapport à cet invariant). Les deux parties reposent largement sur des théorèmes topologiques classiques.

Il est instructif de comparer la preuve originale d'un résultat préexistant avec l'argument déployé à partir de notre cadre. Il est intéressant de constater que, dans certains cas, les arguments sous-jacents sont essentiellement les mêmes, comme dans le cas des ensembles chainables et des pseudo-cubes. Cependant, dans d'autres cas, les arguments sous-jacents diffèrent considérablement, comme dans le cas des variétés compactes. Dans ce scénario, le nouvel argument fournit des informations supplémentaires, notamment une mesure précise de la non-uniformité du calcul (voir le 2). De plus, il implique un résultat supplémentaire : les cônes de variétés sont de type calculable (fort), voir corollaire 5.4.1. Il est important de noter que cela se fait au prix d'une forte dépendance à l'égard des résultats de la topologie algèbrique concernant l'homologie et la cohomologie des variétés, alors que la preuve présentée dans [START_REF] Iljazovic | Compact manifolds with computable boundaries[END_REF] est locale et utilise principalement les propriétés des boules et des sphères, notamment une forme du théorème du point fixe de Brouwer.

La propriété de (l' -)surjection. Nous établissons une condition nécessaire purement topologique pour le type calculable fort, qui est également suffisante pour certains espaces. Cette condition est appelée propriété de l' -surjection. Plus précisément, une paire (X, A) satisfait la propriété de l'surjection si toute fonction continue qui est l'identité sur A et est -proche de l'identité sur X est surjective. Nous démontrons que satisfaire la propriété de l' -surjection pour un certain > 0 est une condition nécessaire pour être de type calculable fort (corollaire 4.4.1). Pour certains espaces, tels que les complexes simpliciaux finis, c'est également suffisant (corollaire 6.7.1).

Il s'avère que la propriété de l' -surjection est liée à ce que nous appelons la propriété de surjection. Une paire (X, A) satisfait la propriété de surjection si toute fonction continue qui est l'identité sur A est surjective. Nous étudions cette notion en détail en utilisant la théorie de l'homologie.

Remarquons que dans un complexe simplicial, chaque sommet a un voisinage qui est un cône, avec l'apex au sommet. Nous démontrons que la propriété de l' -surjection de l'espace est équivalente à la propriété de surjection pour chaque tel cône (voir corollaire 6.7.1). Iljazović et al. ont déjà observé que le type calculable était une propriété locale, voir par exemple [START_REF] Iljazović | Semicomputable manifolds in computable topological spaces[END_REF]. Nous démontrons que pour les complexes simpliciaux finis, le type calculable fort est équivalent au type calculable (voir encore une fois le corollaire 6.7.1).

Nous relions la notion de cycle en homologie à la propriété de surjection pour les cônes, cette notion fournit plusieurs caractérisations (voir la section 6.5.3).

Comme application, nous démontrons que pour les paires simpliciales finies (X, A) telles que X est pure ou de dimension au plus 4, la question de savoir si (X, A) est de type calculable est décidable (voir le corollaire 6.7.3).

Grâce à ces résultats, il devient très facile d'établir si un complexe simplicial de dimension 2 est de type calculable (fort), en examinant certains graphes apparaissant dans le complexe (les liens des sommets) et en déterminant si dans ces graphes, chaque arête appartient à un cycle. Comme application, nous démontrons par exemple que la maison de Bing est de type calculable (fort) xvii (proposition 6.7.2), voir la Figure 2. En outre, nous montrons que la propriété de type calculable n'est pas préservée par les produits (théorème 6.6.2). Cela résout une question de Čelar et Iljazović et est dû au comportement sophistiqué de l'homomorphisme de suspension entre les groupes d'homotopie des sphères. Ainsi, il semble que la résolution de cette question aurait été impossible sans une étude topologique approfondie de la propriété de type calculable.

Complexité descriptive des invariants topologiques. Les résultats mentionnés ci-dessus nous incitent à étudier indépendamment la complexité descriptive des invariants topologiques, notamment les invariants de complexité Σ 0 2 . La théorie descriptive des ensembles est une branche des mathématiques qui étudie la structure et la classification des sous-ensembles d'espaces topologiques en termes de leur définissabillité et xviii de leur complexité. Nous mesurons la complexité descriptive d'un invariant comme la complexité descriptive de la collection des sous-ensembles compacts du cube de Hilbert satisfaisant cet invariant.

Nous donnons une caractérisation complète des invariants de complexité Π 0 1 (théorème 5.5.1), montrant qu'ils ne peuvent exprimer que des propriétés de connexité de l'espace. De plus, la classe non effective des invariants Π 0 1 et la classe effective des invariants Π 0 1 coïncident. Nous menons une étude approfondie de la classe des invariants Σ 0 2 . Nous obtenons une caractérisation indiquant quand un espace peut être séparé d'un autre par un invariant Σ 0 2 (théorème 5.6.1). Nous appliquons ce résultat pour montrer que le segment de droite et la courbe sinus fermée du topologue ne peuvent pas être séparés par des invariants Σ 0 2 (proposition 5.6.2), et nous présentons d'autres applications (théorèmes 5.6.2 et 5. 6.3).

Nous explorons le pouvoir expressif des invariants Σ 0 2 en identifiant les espaces qu'ils peuvent séparer. Les invariants traditionnels ont tendance à présenter une complexité élevée, ce qui nécessite le développement de nouveaux invariants. Pour cela, nous partons des espaces que nous souhaitons distinguer et construisons des invariants de faible complexité adaptés. Nous appliquons cette stratégie à la famille des graphes topologiques finis, caractérisons les couples de graphes pouvant être séparés par un invariant Σ 0 2 , et en déduisons que toute paire de graphes peut être séparée dans un sens ou dans l'autre par de tels invariants (théorèmes 5.7.2 et 5.7.3).

Orientations futures. Notre étude a révélé plusieurs questions intrigantes qui constituent des directions de recherche prometteuses, dont certaines sont mises en évidence dans la section Perspectives.

Par exemple, nos résultats soulèvent la question suivante : si un complexe simplicial fini est de type calculable (fort), est-il minimal par rapport à un invariant topologique Σ 0 2 ? Nos résultats sur la complexité descriptive de séparation des graphes finis font émerger un problème plus général : pour chaque entier n, quel niveau de complexité descriptive des invariants topologiques permet de séparer toute paire de complexes simpliciaux finis de dimension n? Dans le cas des graphes à une dimension, existe-t-il une classe "naturelle" d'invariants Σ 0 2 capable de distinguer les graphes topologiques finis?

De plus, nous réfléchissons à la complexité descriptive impliquée dans la reconnaissance d'espaces fondamentaux tels que les cercles ou les disques. Cette question s'étend à tout espace compact, ce qui nous pousse à étudier la complexité descriptive requise pour leur reconnaissance.

L'exploration de ces questions ouvertes présente de grandes promesses pour faire progresser notre compréhension du sujet et stimuler de nouvelles recherches dans le domaine.

Structure de la thèse Présentons l'organisation de la thèse en donnant le contenu de chaque chapitre.

Dans le chapitre 1, nous donnons un aperçu de nos principales contributions. Dans le chapitre 2, nous présentons les concepts standards de calculabilité sur les espaces topologiques et rappelons la définition du type calculable. Nous proposons un récapitulatif concis des concepts clés et des résultats requis de topologie générale et algébrique dans le chapitre 3. Dans le chapitre 4, nous étudions en détail la notion de type calculable fort. Dans le chapitre 5, nous étudions la complexité descriptive des invariants topologiques. Dans le chapitre 6, nous étudions en détail la propriété de surjection et l'appliquons aux complexes simpliciaux finis.

Dans la section Perspectives, nous dressons une liste succincte des questions ouvertes. Dans les annexes A et B, nous présentons certaines démonstrations que nous n'avons pas trouvées dans la littérature.

xix

Introduction in English

C

omputable analysis provides notions of computability for various mathematical objects, such as real numbers, sets, functions, etc. We focus in particular on compact subsets of Euclidean spaces and more general topological spaces, for which the two significant notions of computable compact sets and semicomputable compact sets are defined. In essence, a subset of the Euclidean plane is considered computable if it is possible to devise a program capable of rendering the set on a screen with arbitrary resolution. On the other hand, a subset is deemed semicomputable if there exists a program that can reject points lying outside the set. A well-known illustration of these concepts is the Mandelbrot set, which can be easily recognized as semicomputable based on its definition. However, determining its computability remains an open problem, tied to a conjecture in complex dynamics [START_REF] Hertling | Is the Mandelbrot set computable?[END_REF].

Miller's pioneer work [START_REF] Miller | Effectiveness for embedded spheres and balls[END_REF] revealed that for certain sets, semicomputability and computability are, astonishingly, equivalent. He proved that finite-dimensional spheres embedded in Euclidean spaces exhibit this remarkable property. Subsequently, Iljazović delved into a systematic exploration of this computability-theoretic attribute, focusing on spheres embedded in computable metric spaces [START_REF] Iljazović | spheres and cells in computable metric spaces[END_REF], on compact manifolds [START_REF] Iljazovic | Compact manifolds with computable boundaries[END_REF], and on various other sets, as evidenced by a series of articles co-authored with several researchers [START_REF] Iljazović | Chainable and circularly chainable co-r.e. sets in computable metric spaces[END_REF][START_REF] Iljazović | spheres and cells in computable metric spaces[END_REF][START_REF] Iljazovic | Compact manifolds with computable boundaries[END_REF][START_REF] Burnik | Computability of 1-manifolds[END_REF][START_REF] Iljazović | Semicomputable manifolds in computable topological spaces[END_REF][START_REF] Čičković | Chainable and circularly chainable semicomputable sets in computable topological spaces[END_REF][START_REF] Horvat | Computability of pseudo-cubes[END_REF][START_REF] Iljazović | Computability of graphs[END_REF][START_REF] Čelar | Computability of products of chainable continua[END_REF][START_REF] Čelar | Computability of glued manifolds[END_REF].

These works have led to the definition of computable type: a compact metrizable space X has computable type if all semicomputable copies of it in the Hilbert cube are computable (the Hilbert cube being an infinite countable Cartesian product of closed intervals [0, 1]). A compact pair (X, A compact with A ⊂ X) is said to have computable type if, for any copy (Y, B) of it, if both Y and B are semicomputable, then Y is computable. Thus, the interval [0, 1] is not computable, but the pair ([0, 1], {0, 1}) is computable.

This property serves as the central focus of investigation within this thesis. Our objective is to undertake a comprehensive theoretical investigation of the computable type notion, aiming to enhance our understanding of this concept and provide a toolkit for proving or disproving this property. We have noticed that the existing techniques in the literature are usually very specific and could benefit from incorporating more unifying arguments, which could be addressed by exploring a fresh approach.

This interdisciplinary project encompasses computable analysis, descriptive set theory, and general and algebraic topology. The examination of the computability property in question intricately intersects with topology, making it essential to incorporate into our investigation.

In this introduction, we will provide a brief and informal overview of the results we have obtained in various directions. Further elaboration and details will be presented in subsequent parts.

The notion of computable type. Mainly two notions of computable type have been introduced in the literature, that consider copies of the space embedded in different classes of topological spaces, namely computable metric spaces and computably Hausdorff spaces. Several results have been established on computable metric spaces first [START_REF] Iljazović | Chainable and circularly chainable co-r.e. sets in computable metric spaces[END_REF][START_REF] Iljazovic | Compact manifolds with computable boundaries[END_REF] and then extended to computably Hausdorff spaces [START_REF] Čičković | Chainable and circularly chainable semicomputable sets in computable topological spaces[END_REF][START_REF] Iljazović | Semicomputable manifolds in computable topological spaces[END_REF]. We first show that these two notions are actually equivalent, and it is sufficient to consider copies in the Hilbert cube (see Theorem 4.2.1).

The notion of strong computable type. Obtaining comprehensive general characterizations of the computable type property is not possible since most spaces trivially have computable type by lacking semicomputable copies. To overcome this limitation we introduce a more flexible notion that we call strong computable type which is having computable type relative to any oracle (see Definition 4.3.1).

xxi With the notion of strong computable type we are able to obtain many topological characterizations and results (see Theorems 4.3.1 and 4.3.3). Furthermore, it reveals close connections to other notions that motivate independent exploration and further development (see Corollary 4.4.3). In the next two paragraphs we give some details about these results.

Strong computable type and minimality. Our first characterization is that a space has strong computable type if and only if it satisfies a property of low descriptive complexity, and is minimal in the sense that no proper subspace satisfies this property (Theorems 4.3.1 and 4.3.3). In particular, every space which is minimal satisfying some Σ 0 2 invariant has strong computable type (Theorem 4.3.2). It serves as a motivation to search for Σ 0 2 invariants, allowing to revisit previous results and obtain new ones. This finding significantly expands our understanding and provides a common template for unifying all the proofs of the strong computable type property.

We identify classes of Σ 0 2 invariants involving continuous functions to spheres, and more generally compact absolute neighborhood retracts (ANRs), see Section 5.2.3. When Y is a fixed compact ANR and X is a varying compact space, we show how the set [X; Y ] of continuous functions from X to Y quotiented by homotopy can be computed from X, with no computability assumptions about Y (Theorem 5.2.1). A particularly interesting case is when Y is the n-dimensional sphere S n . This analysis immediately induces topological invariants of low descriptive complexity, which are very classical in topology. We define the invariants E n and H n that capture the extensibility and nullhomotopy of continuous functions to the n-sphere S n (see Definitions 5.3.2 and 5.3.3).

We thoroughly study these topological invariants and revisit several results of the literature about computable type by identifying a suitable Σ 0 2 invariant for which the space or the pair is minimal (see Section 5.4). These results show that the theory indeed applies, and they strengthen the previous results because they give more information and can be used to derive new results. For instance,

The pair (B n+1 , S n ) is E n -minimal (Proposition 5.3.6), Every closed n-manifold is H n -minimal (Theorem 5.4.1),

The pair consisting of the Warsaw saucer with its bounding circle is E 1 -minimal and hence has strong computable type (Proposition 5.4.3), see Figure 5.

Our framework often yields simpler proofs of results by splitting the argument into two parts: a computability-theoretic component (demonstrating that a topological invariant is Σ 0 2 ) and a purely topological component (establishing that a space is minimal with respect to that invariant). Both parts heavily rely on classical topological theorems.

It is instructive to compare the original proof of a previous result with the unfolded argument derived from our framework. Interestingly, we find that in some cases, the underlying arguments are essentially the same, such as in the case of chainable sets and pseudo-cubes. However, for other cases, the underlying arguments differ significantly, as seen in the case of compact manifolds. In this scenario, the new argument provides additional information, including a precise measure of the non-uniformity of the computation (see Theorem 4.4.2). Moreover, it implies an additional result: cones of manifolds have (strong) computable type (Corollary 5.4.1). It is important to note that this comes at the cost of relying heavily on algebraic topology results concerning the homology and cohomology of manifolds, whereas the proof presented in [START_REF] Iljazovic | Compact manifolds with computable boundaries[END_REF] is local and primarily utilizes properties of balls and spheres, notably a form of Brouwer's fixed-point theorem.

The ( -)surjection property. We establish a purely topological necessary condition for strong computable type, which is also sufficient for some spaces. This condition is called the -surjection property. Namely, a pair (X, A) satisfies the -surjection property if every continuous function xxii which is the identity on A and is -close to the identity on X is surjective. We prove that satisfying the -surjection property for some > 0 is a necessary condition to have strong computable type (Corollary 4.4.1). For certains spaces, such as finite simplicial complexes, it is also sufficient (Corollary 6.7.1).

It turns out that -surjection property is related to what we call the surjection property. A pair (X, A) satisfies the surjection property if every continuous function which is the identity on A is surjective. We study this notion in details using homology theory. Note that in a simplicial complex, each vertex has a neighborhood which is a cone, with the tip at the vertex. We prove that the -surjection property is equivalent to the surjection property for each such cone (see Corollary 6.7.1). Iljazović et al. already observed that computable type was a local property see for instance [START_REF] Iljazović | Semicomputable manifolds in computable topological spaces[END_REF]. We prove that for finite simplicial complexes, strong computable type is equivalent to computable type (again see Corollary 6.7.1).

We relate the notion of cycle from homology to the surjection property for cone pairs., this notion provide several characterizations (see Section 6.5.3).

As an application, we prove that for finite simplicial pairs (X, A) such that X is pure or has dimension at most 4, whether (X, A) has computable type is decidable (see Corollary 6.7.3).

With these results, it becomes very easy to establish whether a 2-dimensional simplicial complex has (strong) computable type, by inspecting certain graphs appearing in the complex (the links of the vertices), and determining whether in these graphs, each edge belongs to a cycle. As an application we prove for instance that Bing's house has (strong) computable type (Proposition 6.7.2), see Figure 6. 8: A semicomputable copy of a disk attached to a pinched torus which is not computable Furthermore, we show that the computable type property is not preserved by products (Theorem 6.6.2). This solves a question by Čelar and Iljazović and is due to the sophisticated behavior of the suspension homomorphism between the homotopy groups of spheres. Thus, it appears that the resolution of this question would have been impossible without a thorough topological study of the computable type property.

Descriptive complexity of topological invariants. The above mentioned results motivate to study independently the descriptive complexity of topological invariants, notably the invariants of complexity Σ 0 2 . Descriptive Set Theory is a branch of mathematics that investigates the structure and classification of subsets of topological spaces in terms of their definability and complexity. We measure the descriptive complexity of an invariant as the descriptive complexity of the collection of compact subsets of the Hilbert cube satisfying this invariant.

xxiv

We give a complete characterization of the invariants of complexity Π 0 1 (Theorem 5.5.1), showing that they can only express connectedness properties of the space. Moreover, the non-effective class of Π 0 1 invariants and the effective class of Π 0 1 invariants coincide. We carry out a thorough study of the class of Σ 0 2 invariants. We obtain a characterization of when a space can be separated from another by some Σ 0 2 invariant (Theorem 5.6.1). We apply this result to show that the line segment and the closed topologist's sine curve cannot be separated by Σ 0 2 invariants (Proposition 5.6.2), and give other applications (Theorems 5.6.2 and 5.6.3). We explore the expressive power of Σ 0 2 invariants by identifying which spaces they can separate. Traditional invariants tend to exhibit high complexity, necessitating the development of novel invariants. To achieve this, we initiate the process by considering the spaces we aim to distinguish and we construct suitable low complexity invariants. We apply this strategy to the family of finite topological graphs, characterize pairs of graphs that can be separated by a Σ 0 2 invariant, and deduce that any pair of graphs can be separated in one way or another by such invariants (Theorems 5.7.2 and 5.7.3).

Future directions. Our study has unveiled several intriguing questions that serve as promising research directions, some of which are highlighted in Section Perspectives.

For instance, our results raise the following question: if a finite simplicial complex has (strong) computable type, is it minimal with respect to a Σ 0 2 topological invariant? Our results on the descriptive complexity of separating finite graphs highlight a broader problem: for each integer n, what level of descriptive complexity of topological invariants allows us to separate any pair of finite simplicial complexes of dimension n? In the case of 1-dimensional graphs, is there a "natural" class of Σ 0 2 invariants capable of discerning finite topological graphs? Additionally, we ponder over the descriptive complexity involved in recognizing fundamental spaces such as circles or disks. This question extends to any compact space, prompting us to investigate the descriptive complexity required for their recognition.

The exploration of these open questions holds great promise for advancing our understanding of the subject matter and stimulating further research in the field.

Thesis Structure. Let us show how the thesis is organized by giving the content of each chapter.

In Chapter 1 we give an overview of our main contributions. In Chapter 2, we present the standard concepts of computability over topological spaces and recall the definition of computable type. A concise recapitulation of the key concepts and necessary results from general and algebraic topology is provided in Chapter 3. In Chapter 4, we study in details the notion of strong computable type. In Chapter 5, we study the descriptive complexity of topological invariants. In Chapter 6, we study in details the surjection property and apply to finite simplicial complexes.

In Section Perspectives, we give a brief list of open questions. In Appendixes A and B we give some proofs that we did not find in the literature.

xxv

List of Papers

Published

Strong Computable Type, Computability, 2023, with Mathieu Hoyrup, see [START_REF] Djamel | Strong computable type. Computability[END_REF].

Comparing Computability in two topologies, Journal of Symbolic Logic, 2023, with Mathieu Hoyrup, see [6],

Computability of Finite Simplicial Complexes, Leibniz International Proceedings in Informatics (LIPIcs) 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022), with Mathieu Hoyrup, see [START_REF] Djamel | Computability of finite simplicial complexes[END_REF].

Submitted

Descriptive Complexity of Topological Invariants, 2023, with Mathieu Hoyrup, see [START_REF] Djamel | Descriptive complexity of topological invariants[END_REF].

The Surjection Property and Computable Type, 2023, with Mathieu Hoyrup, see [START_REF] Djamel | The surjection property and computable type[END_REF].

xxvii Chapter 1

Overview

In this chapter, we present a summary of our main contributions, focusing on those that can be easily expressed.

The contributions encompass the introduction of new notions, the development of characterizations, the formulation of theorems and the provision of (counter)examples.

By highlighting these key elements, we aim to provide a concise overview of the significant outcomes and areas of inquiry within our research.

The reader is invited to read the introduction before reading this overview. Let Q = [0, 1] N be the Hilbert cube endowed with the product topology, induced by the complete metric d Q (x, y) = i 2 -i |x i -y i | where x = (x 0 , x 1 , . . .) and y = (y 0 , y 1 , . . .).

Theorem. (Theorem 4.2.1) Computable type can be equivalently defined on the Hilbert cube, on computable metric spaces and on computably Hausdorff spaces.

The proof is based on Schröder's effective metrization theorem [START_REF] Schröder | Effective metrization of regular spaces[END_REF] which implies that compact Hausdorff spaces are metrizable in an effective way.

Strong Computable Type

We introduce a more flexible notion relativizing the notion of computable type.

Let us give some central definitions. A compact pair (X, A) consists of a compact metrizable space X and a compact subset A ⊆ X. A copy of a pair (X, A) in a topological space Z is a pair (Y, B) such that Y ⊆ Z is homeomorphic to X and the image of A by the homeomorphism is B.

Definition. (Definition 4.3.1) A compact pair (X, A) has strong computable type if for every oracle O and every copy (Y, B) of (X,

A) in Q, if (Y, B) is semicomputable relative to O, then Y is computable relative to O.
We study in details this notion in Chapter 4 and Section 5.4.

Relative Strong Computable Type

We prove that the relativization of strong computable type is captured by a purely topological property which involves -deformations.

Let d H be the induced Hausdorff distance on the set of non-empty compact subsets of Q.

Definition. (Definition 4.3.4) Let X ⊆ Q and > 0. An -function is a continuous function f : X → Q such that d(f (x), x) < for all x ∈ X. An -deformation of X is the image of X under an -function.

Corollary. (Corollary 4.4.2) For a compact pair (X, A), the following statements are equivalent:

1. (X, A) has relative strong computable type (see Definition 4.4.2), 2. There exists > 0 such that no sequence of -deformations of (X, A) converges to a proper compact subpair of (X, A) in the Hausdorff distance.

In particular, if a compact pair (X, A) has strong computable type then it satisfies 2. (see Corollary 4.4.1).

Computable Type vs Topological Invariants

Let K(Q) be the hyperspace of Q, we fix a topology on K(Q), which is either the Vietoris or the upper Vietoris topology. The descriptive complexity of a property P ⊆ K(Q) in that topology is defined as follows.

It is Σ 0 1 if it is open, it is Π 0 1 if it is closed, it is Σ 0 2 if it is a countable union of differences of Π 0
1 sets (equivalently, a countable uninon of Π 0 1 sets when K(Q) is endowed with the Vietoris topology).

Effective complexity classes Π 0 1 and Σ 0 2 can be defined by requiring that the corresponding description of the set can be produced by a program.

1) Σ 0 2 Invariants and Strong Computable Type

The next result gives a relatively simple recipe to prove that a space (or pair) has strong computable type.

Theorem. (Theorem 4.3.2) If a compact pair (X, A) is minimal satisfying some Σ 0 2 topological invariant (in Vietoris or upper Vietoris topology), then (X, A) has strong computable type.

Note that this theorem is not an equivalence (see Section 5.4.4 for a counter-example). Being minimal for some Σ 0 2 invariant is therefore a distinguished way of having strong computable type, from which particular consequences can be derived, for instance about the level of uniformity of the computation that is involved in strong computable type (see Section 4) below).

2) The invariants E n and H n

We define two families of Σ 0 2 invariants E n and H n which play a key role in our study of strong computable type. Namely, for every n ∈ N, we define the invariants E n and H n . Let S n be the ndimensional sphere.

Definition. (Definition 5.3.2) For n ∈ N, a compact pair (X, A) is in E n if there exists a continuous function f : A → S n that has no continuous extension F : X → S n .

Definition. (Definition 5.3.3) A compact pair (X, A) is in H n if there exists a continuous function of pairs f : (X, A) → (S n , p) which is not null-homotopic relative to A (i.e. the homotopy is constant on A).

These invariants are studied in Sections 5.2, 5.3 and 5.4. They have low descriptive complexity.

Theorem. (Corollaries 5.2.1 and 5.3.3) The topological invariants E n and H n are Σ 0 2 in the upper Vietoris topology.

3) Some Applications

We apply this theory by revisiting the previous results and obtain new ones.

Example. (Theorem 5.4.1) Every connected compact n-manifold M is H n -minimal. Every connected compact n-manifold with boundary (M, ∂M ) is H n -minimal. It gives an alternative proof that they have strong computable type [START_REF] Iljazović | Chainable and circularly chainable co-r.e. sets in computable metric spaces[END_REF][START_REF] Čičković | Chainable and circularly chainable semicomputable sets in computable topological spaces[END_REF][START_REF] Horvat | Computability of pseudo-cubes[END_REF].

Example. (Propositions 5.4.1 and 5.4.2) If X is compact connected and is chainable from a to b, then the pair (X, {a, b}) is E 0 -minimal.

Every pseudo-n-cube is E n-1 -minimal.

It gives an alternative proof that they have strong computable type [START_REF] Iljazović | Semicomputable manifolds in computable topological spaces[END_REF].

Furthermore, we are able to obtain new results effortlessly or at no additional cost, thanks to our findings.

Example. (Corollary 5.4.1) If M is a compact connected n-manifold with possibly empty boundary ∂M , then the pair

C(M, ∂M ) = (C(M ), M ∪ C(∂M ))
is E n -minimal hence has strong computable type.

Example. (Proposition 5.4.

3) The pair consisting of the Warsaw saucer with its bounding circle is E 1 -minimal and hence has strong computable type (see Figure 5).

4) Uniformity

Typically, in cases where a pair has computable type, there is a lack of a uniform method for transforming semicomputability into computability. We focus on examining the non-uniformity of computation by employing Weihrauch degrees.

Definition. (Definition 4.4.4) For a compact pair (X, A), let SCT (X,A) be the function taking a copy (Y, B) of (X, A) in τ 2 upV and outputting Y in τ 2 V .

Definition. (Definition 4.4.5) Closed choice over N is the problem C N of finding an element in a non-empty set A of natural numbers, given any enumeration of the complement of A.

Strong Weihrauch reducibility relative to some oracle is usually denoted by ≤ t sW (where t stands for topological ).

The first result holds for every compact pair (which may or may not have strong computable type).

Theorem. (Theorem 4.4.1) Let (X, A) be a compact pair such that X is not a singleton. One has C N ≤ t sW SCT (X,A) . If (X, A) has a semicomputable copy then C N ≤ sW SCT (X,A) .

If the pair is minimal for some Σ 0 2 invariant, then the non-uniformity level of the computation is precisely what is called non-deterministic computability, captured by the Weihrauch degree of choice over N (Theorem 4.4.2).

Theorem. (Theorem 4.4.2) Let P be a Σ 0 2 invariant. If a compact pair (X, A) is P-minimal then SCT (X,A) ≤ W C N .

The -Surjection Property

We introduce a necessary condition for strong computable type, which is sufficient for certain classes of spaces.

Definition. (Definition 4.4.3) Let > 0. A pair (X, A) ⊆ Q satisfies the -surjection property if every continuous function f : X → X satisfying d X (f, id X ) < and f | A = id A is surjective.
The pair satisfies the surjection property if every continuous function f :

X → X such that f | A = id A is surjective.
It turns out that the ( -)surjection property is very related to the notion of strong computable type. We study it in details in Chapter 6 and a bit in Section 4.4.

A Necessary Condition to have Strong Computable Type

We prove that the -surjection property is necessary to have strong computable type.

Corollary. (Corollary 4.4.3) If (X, A) ⊆ Q has strong computable type, then it satisfies the -surjection property for some > 0.

Example. (Proposition 4.4.3) If X contains infinitely many isolated points, then X does not have strong computable type.

Failing to have Computable Type

1) Computable Witnesses

We have seen that if a compact pair does not satisfy the -surjection property for any > 0, then it does not have strong computable type (Corollary 1.1.4).

We introduce an effective version of "not satisfying the -surjection property for any > 0" and prove that it implies that the pair does not have computable type (what we prove is slightly stronger, because we consider a stronger version of the -surjection property). We study it in Section 4.5.1.

Definition. (Definition 4.5.4) Let (X, A) be a compact pair in Q. For > 0, say that δ > 0 is an -witness if there exists a continuous function f :

X → X satisfying f (A) ⊆ A and d X (f, id X ) < , such that d H (X, f (X)) > δ.
Say that (X, A) has computable witnesses if there exists a computable function sending each rational > 0 to a rational -witness δ > 0.

Theorem. (Theorem 4.5.2) Let (X, A) ⊆ Q be a pair of semicomputable compact sets. If it has computable witnesses, then (X, A) does not have computable type, i.e. there exists a semicomputable copy of (X, A) such that the copy of X is not computable.

2) Spaces without proper copies of themselves

We explain why for some spaces it is difficult to produce a semicomputable copy which is not computable (Theorem 4.5.3), contrasting with obvious examples such as the line segment or the ndimensional ball.

Example. (Theorem 4.5.3) If X is computable and does not properly contain a copy of itself, then there is no geometrical transformation (scaling, rotation, translation) yielding a semicomputable copy of X which is not computable, and more generally there is no bilipschitz transformation yielding such a copy.

Descriptive Complexity of Topological Invariants

The insightful connection between strong computable type and topological invariants serves as a compelling motivation to conduct an independent study on the descriptive complexity of these invariants.

One approach to establishing strong computable type involves identifying a Σ 0 2 invariant for which the space/pair is minimal. Consequently, a crucial endeavor arises: the search for an extensive collection of Σ 0 2 invariants. This pursuit leads us to the overarching challenge of determining the range of invariants expressible within a given limit of complexity. To embark on this exploration, we initially focus on Π 0 1 invariants as a starting point.

Characterization of the Π 0 1 Topological Invariants

We define a class of Π 0 1 topological invariants.

Definition. (Definition 5.5.1) Let 0 ≤ p ≤ n be natural numbers. The topological invariant C n,p is defined by: X ∈ C n,p iff X has at most n connected components, among which at most p non-trivial ones.

In particular, X ∈ C n,n iff X has at most n connected components; X ∈ C n,0 iff X contains at most n points.

In Section 5.5 we show how these invariants generate all Π 0 1 invariants.

Theorem. (Theorem 5.5.1) Let P be a non-trivial topological invariant. The following statements are equivalent:

1. P ∈ Π 0 1 , 2. P ∈ Π 0 1 , 3. P is a finite union of C n,p 's.
1.2.2 The Σ 0 2 Invariants 1) Strong Approximation and Σ 0 2 Invariants A way to investigate the expressive power of Σ 0 2 invariants is to understand which spaces can be separated by such invariants. We first give a characterization of the separability of two spaces.

We say that a compact space X strongly approximates a compact space Y if there is a copy X 0 ⊆ Q of X such that for every > 0, one can continuously deform X 0 to converge to a copy of Y , in such a way that every point of X 0 is moved by at most . Definition. (Definition 5.6.1) Let X, Y be compact spaces. We say that X strongly approximates Y , written X Y , if there exists a copy X 0 ⊆ Q of X such that for every > 0, some copy of Y is a limit in the Hausdorff distance (Vietoris topology) of -deformations of X 0 .

Note that if X strongly approximates Y , then the condition actually holds for every copy X 0 ⊆ Q of X (see Remark 5.6.1).

This notion and its relation with Σ 0 2 and Σ 0 2 invariants is studied in Section 5.6. We obtain the following characterization.

Theorem. (Theorem 5.6.1) Let X, Y be compact spaces. The following statements are equivalent:

X strongly approximates Y , Every Σ 0 2 invariant satisfied by X is satisfied by Y .

2) Some Applications

Example. (Proposition 5.6.

2) The closed topologist's sine curve S = {(x, sin(1/x)) : x ∈ (0, 1]} ∪ {(0, y) : y ∈ [-1, 1]} and the line segment cannot be distinguished by Σ 0 2 invariants.

Example. (Example 5.7.1) Every Σ 0 2 invariant satisfied by the line segment is satisfied by the graphs that have a bridge, i.e. an edge whose removal disconnects the graph. Every Σ 0 2 invariant satisfied by the circle is satisfied by the graphs that have a cycle. Moreover, the line segment and the circle can be separated by some Σ 0 2 invariant, in both directions.

Separating Finite Topological Graphs

We study the relation between strong approximation and graphs (see Section 5.7). We show that for finite topological graphs, the non-effective class Σ 0 2 is no more expressive than the effective class Σ 0 2 .

Theorem. (Theorem 5.7.1) Let X be a finite topological graph and Y a compact space. If there exists a Σ 0 2 invariant satisfied by X but not by Y , then there exists a Σ 0 2 such invariant.

We show that the Σ 0 2 invariants are expressive enough to separate non-homeomorphic graphs.

Theorem. (Theorem 5.7.

3) The following are equivalent G, H are non-homeomorphic finite topological graphs, Some Σ 0 2 invariant is satisfied by G but not by H, or vice-versa.

Note that there is a dissymmetry: it can happen that every Σ 0 2 invariant satisfied by G is also satisfied by H (but not vice-versa).

Let G and H be finite topological graphs, we give a characterization of the following relation: every Σ 0 2 invariant is satisfied by G but not by H. We say that H can be contracted to G if there is a sequence of edge contractions from H to G; an edge contraction consists in removing an edge and merging its two endpoints (we actually consider multigraphs with loops).

Theorem. (Theorem 5.7.2) The following statements are equivalent for finite connected graphs G, H: Every Σ 0 2 invariant satisfied by G is satisfied by H, Some subdivision of H can be contracted to G.

The Surjection Property

In this section we study more in details the ( -)surjection property (see Definition 1.1.4) and its relation with (strong) computable type.

The -surjection property does not depend on the choice of a compatible metric on X (see Proposition 6.3.1)

We use the notion of cones and cone pairs.

Definition. (Definitions 3.2.2 and 3.2.3) Let X be a topological space. The cone of X is the quotient of X × [0, 1] under the equivalence relation (x, 0) ∼ (x , 0). Let (X, A) be a pair. Its cone pair is the pair C(X, A) = (C(X), X ∪ C(A)).

Note that for a compact pair (X, A), the cone pair C(X, A) has the -surjection property for some > 0 iff it has the surjection property (Proposition 6.3.3).

Let X be a topological space and n ∈ N,

n ≥ 1. A point x ∈ X is n-Euclidean if it has an open neighborhood that is homeomorphic to R n . A point
A set C ⊆ X is a regular n-cell if there is a homeomorphism f : B n → C such that f (B n \ S n-1
) is an open subset of X. A regular cell is a regular n-cell for some n ≥ 1.

The ( -)Surjection Property and Unions

We reduce the ( -)surjection property for a pair to the ( -)surjection property for subpairs covering it.

Theorem. (Theorem 6.3.1) Let (X, A) and (X i , A i ) i∈N be compact pairs such that X = i∈N X i and A = i∈N A i . Assume that (X, A) is almost Euclidean. If every pair C(X i , A i ) has the surjection property, then C(X, A) has the surjection property.

Theorem. (Theorem 6.3.2) Let (X, A) and (X i , A i ) i≤n be compact pairs such that X = i≤n X i and A = i≤n A i . Assume that each topological boundary ∂X i is a neighborhood retract in X. If every pair (X i , A i ) has the -surjection property for some > 0, then (X, A) has the δsurjection property for some δ > 0.

Finitely Conical Spaces

For the next class of spaces we prove that the -surjection property is actually a local property.

Definition. (Definition 6.4.1) A topological space X is finitely conical if there exists a finite sequence of compact metrizable spaces (L i ) i≤n and a finite covering of X by open sets U i ⊆ X, i ≤ n, where U i is homeomorphic to OC(L i ). A pair (X, A) is finitely conical if X is finitely conical and for every i, the homeomorphism f i :

U i → OC(L i ) satisfies f (U i ∩ A) = OC(N i ) for some compact N i ⊆ L i .
Note that finitely conical pairs are closed under finite products and the cone operator (see Proposition 6.4.1).

Theorem. (Theorem 6.4.1)

Let (X, A) be a finitely conical compact pair coming with (L i , N i ) i≤n , where each L i is an ANR. The following statements are equivalent:

1. (X, A) has the -surjection property for some > 0, 2. All the cone pairs C(L i , N i ) have the surjection property.

The Surjection Property for Cone Pairs

1) Quotients and Retractions to Spheres

The surjection property for cone pairs is intimately related to quotients and retractions to spheres.

Theorem. (Theorem 6.5.1) Let (X, A) be an almost Euclidean compact pair. The following statements are equivalent:

1. C(X, A) has the surjection property, 2. For every n ≥ 1 and every regular n-cell C ⊆ X \ A, the quotient map q C : (X, A) → (S n , s) is not null-homotopic.

Corollary. (Corollary 6.5.1) Let (X, A) be an almost Euclidean compact pair. The pair C(X, A) has the surjection property if and only if C(X/A, ∅) = (C(X/A), X/A) has the surjection property.

Theorem. (Theorem 6.5.2) Let (X, A) be a compact pair and C ⊆ X \A be a regular n-cell.

If there exists a retraction r : X \ op(C) → bd(C) which is constant on A, then the quotient map q : (X, A) → (S n , s) is null-homotopic, hence C(X, A) does not have the surjection property.

We prove that condition that r| A is constant can be replaced by being null-homotopic (Proposition 6.5.2).

2) Cycles

Let T = R/Z is the circle group.

Definition. (Definition 6.5.1) Let (X, A) be a compact pair and n ≥ 1. A regular ncell C ⊆ X \ A belongs to a relative cycle if the canonical homomorphism

H n (X, A; T) → H n (X, X \ op(C); T) is non-trivial.
For ANRs, this notion can equivalently be reformulated using cohomology (see Remark 6.5.1). For finite simplicial complexes, it can be reformulated using maximal simplices (see Proposition 6.5.3).

Note that H n (X, X \ op(C); T) ∼ = T, that is why we obtain the following.

Corollary. (Corollary 6.5.3) Let (X, A) be an almost Euclidean compact pair. If every regular cell C ⊆ X \ A belongs to a relative cycle, then C(X, A) has the surjection property.

Conditions on the dimension give us some characterizations.

Corollary. (Corollary 6.5.4) Let n ≥ 1. Let (X, A) be an almost n-Euclidean compact pair, where X and A are ANRs and dim(X) = n. The following statements are equivalent:

The pair C(X, A) has the surjection property, Every regular cell C ⊆ X \ A belongs to a relative cycle.

Corollary. (Corollary 6.5.5) Let (X, A) be a compact pair, where X and A are ANRs and dim(X) = n ≥ 1. For a regular n-cell C ⊆ X \A, the following statements are equivalent:

C does not belong to a relative cycle,

There exists a retraction r : X \ op(C) → bd(C) which is constant on A.

The equivalence holds if n = 1 or 2, without any dimension assumption about X.

Corollary. (Corollary 6.5.6) Let (X, A) be an almost Euclidean compact pair, where X and A are ANRs and dim(X) ≤ 3. The following statements are equivalent:

The pair C(X, A) has the surjection property, Every regular cell C ⊆ X \ A belongs to a relative cycle.

We also obtain a relationship between cycles and the -surjection property for ANRs.

Theorem. (Theorem 6.5.3) Let (X, A) be an almost Euclidean compact pair, where X is an ANR. If every regular cell C ⊆ X \ A belongs to a relative cycle, then (X, A) has the -surjection property for some > 0.

The product of two pairs (X 1 , A 1 ) and (X 1 , A 1 ) is the pair (

X 1 × X 2 , (X 1 × A 2 ) ∪ (A 1 × X 2 )
). For a finite simplicial pair, if the product satisfies the -surjection property for some > 0, then both (X 1 , A 1 ) and (X 2 , A 2 ) satisfy the δ-surjection property for some δ > 0 (Proposition 6.3.2).

As an application of our results about cycles, we prove surprisingly that the converse does not hold, see Theorem 6.6.2.

Applications to Computable Type

1) A Characterization for Finite Simplicial Complexes

Putting all the results together, we obtain the following characterization for finite simplicial pairs. Corollary 1.3.1. (Corollary 6.7.1) For a finite simplicial pair (X, A) such that A has empty interior in X, the following statements are equivalent:

1. (X, A) has (strong) computable type, 2. (X, A) satisfies the -surjection property for some > 0, 3. For every vertex, the cone pair corresponding to its star has the surjection property (see Section 3.5).

A finite simplicial pair which is a cone, has (strong) computable type iff it satisfies the surjection property.

Using our previous results we obtain the following.

Corollary. (Corollary 6.7.2) Let (X, A) be a finite simplicial pair and (X i , A i ) i≤n be pairs of subcomplexes such that X = i≤n X i and A = i≤n A i . If every pair (X i , A i ) has computable type, then (X, A) has computable type.

Corollary. (Corollary 6.7.3) For finite simplicial pairs (X, A) such that X is pure or has dimension at most 4, whether (X, A) has computable type is decidable.

2) Finite 2-Dimensional Simplicial Complexes

In particular, we obtain a visual way to prove that a finite 2-dimensional simplicial complex has computable type since it is locally a cone of a finite graph.

Corollary. (Corollary 6.7.4) Let (L, N ) be a pair such that L is a finite graph and N is a subset of its vertices. The following statements are equivalent:

1. C(L, N ) has the surjection property, 2. Every edge is in a cycle or a path starting and ending in N , 3. C(L, N ) has computable type.

We apply this to prove that Bing's house has computable type whereas the dunce hat does not, see Propositions 6.7.2 and 6.7.1, refer to the introduction for to see their figures.

3) Boundaries and the Odd Subcomplex

Let ∂ 1 X be the union of simplices that are contained in exactly one simplex of the next dimension, i.e. ∂ 1 X is the union of the free simplices of X, ∂ + X be the union of simplices that are contained in at least one simplex of the next dimension, ∂ odd X be the union of simplices that are contained in an odd number of maximal simplices of the next dimension.

The following observations can be made: Although (X, ∂ 1 X) has computable type when X is a 1-dimensional complex (i.e., a graph), it is no more true for 2-dimensional complexes.

While (X, ∂ + X) always has computable type by Proposition 6.7.3, ∂ + X is far from optimal.

(X, ∂ odd X) always has computable type but ∂ odd X is in general not optimal.

4) Computable Type is Not Preserved by Products

This is a negative answer of an open question, raised by Čelar and Iljazović in [START_REF] Čelar | Computability of products of chainable continua[END_REF]. Note that (B 1 , S 0 ) has computable type.

Corollary 1.3.2. (Corollary 6.7.5) There exists a finite simplicial pair (X, A) that has computable type, but such that the product (X, A) × (B 1 , S 0 ) does not.

Part

W

e give some background about computable aspects of topological spaces, most of which can be found in [START_REF] Weihrauch | Computable Analysis[END_REF][START_REF] Schröder | Admissibly represented spaces and Qcb-spaces[END_REF][START_REF] Iljazović | Computability of subsets of metric spaces[END_REF][START_REF] Kechris | Classical Descriptive Set Theory[END_REF], namely: effective countably-based T 0 -spaces, computable separation axioms, computability of subsets, the Hilbert cube as a universal computable metric space, computable type and descriptive complexity.

First of all, let us recall enumeration reducibility, which enables one to define a notion of computable reduction between points of countably-based topological spaces.

We will mainly use the following notion from computability theory: a set A ⊆ N is computably enumerable (c.e.) if there exists a Turing machine that, on input n ∈ N, halts if and only if n ∈ A. An enumeration of A is any function

f : N → N such that A = {n ∈ N : ∃p ∈ N, f (p) = n + 1}.
A set is c.e. if and only if it has a computable enumeration. This notion immediately extends to subsets of countable sets, whose elements can be encoded by natural numbers such as the set of rational numbers, or the set N * of finite sequences of natural numbers.

The halting set H ⊆ N, which is the set of indices of Turing machines that halt, is a famous example of a c.e. set that is not computable, discovered by Turing in his seminal article [START_REF] Turing | On computable numbers, with an application to the entscheidungsproblem[END_REF].

As we will see soon, a point of a countably-based topological space can be identified with a set of natural numbers, namely the set of indices of its basic neighborhoods. Therefore, relative computability between points will be conveniently expressed using enumeration reducibility, which we recall now.

Let x, D denote the encoding of the pair of finite objects x and D into a natural number. The equivalence between 2. and 3. is due to Selman [START_REF] Selman | Arithmetical reducibilities I[END_REF]. Now, we start our preliminaries about computable analysis.

2.1 Effective countably-based T 0 -spaces Definition 2.1.1. An effective countably-based T 0 -space is a tuple (X, τ, (B i ) i∈N ) where (X, τ ) is a countably-based T 0 topological space and (B i ) i∈N is a numbered basis of τ such that there exists a c.e. set E ⊆ N 3 such that for every i, j ∈ N,

B i ∩ B j = k:(i,j,k)∈E B k . We will say that a point x ∈ X is τ -computable if the set {i : x ∈ B i } is c.e..
We will often denote an effective countably-based T 0 -space by (X, τ ), the numbered basis being implicit. When several topological spaces are involved, we write B X i for the basis of the space X. A subset Y of an effective countably-based T 0 -space (X, τ ) is also an effective countably-based T 0space, by taking the subspace topology and the numbered basis

B Y i = Y ∩ B X i .
The product of two effective countably-based T 0 -spaces is naturally an effective countably-based T 0 -space. In an effective countably-based T 0 -space (X, τ ), a point can be identified with the set of its basic neighborhoods, therefore the computability properties of a point of such a space is entirely captured by the enumeration degree of its neighborhood basis. This idea is explored in depth by Kihara and Pauly in [START_REF] Kihara | Point degree spectra of represented spaces[END_REF]. In particular, a computable reduction between points can be defined using enumeration reducibility as follows.

Definition 2.1.2. Let (X, τ X ) and (Y, τ Y ) be an effective countably-based T 0 -spaces.

A point x in (X, τ X ) is computable relative to a point y in (Y, τ Y ) if i ∈ N : x ∈ B X i ≤ e i ∈ N : y ∈ B Y i .
A function f : X → Y between effective countably-based T 0 -spaces is computable if and only if for every x ∈ X, f (x) is computable relative to x in a uniform way, i.e. using the same machine in the reduction. Equivalently, f :

X → Y is computable if the sets f -1 (B Y i ) are effectively open, uniformly in i.
Let us recall some notions of computability of sets.

Definition 2.1.3. A set A in an effective countably-based T 0 -space (X, τ, (B i ) i∈N ) is:

1. Effectively compact, or semicomputable, if it is compact and the set We will often use the word semicomputable when talking about a subset of a space, and effectively compact when talking about the space itself.

{(i 1 , . . . , i n ) ∈ N * : A ⊆ B i 1 ∪ . . . ∪ B in } is c.e., 2
A compact set is computable iff it is semicomputable and contains a dense computable sequence.

The image of a (semi)computable compact set under a computable function is a (semi)computable compact set. The Mandelbrot set a semicomputable compact set, it is an open question whether it is computable (see [START_REF] Hertling | Is the Mandelbrot set computable?[END_REF]).

The next result is simple but very powerful and central in many arguments: closed sets are preserved by taking existential quantification over a compact set, and it holds effectively. Equivalently, open sets are preserved by taking universal quantification over a compact set, effectively so. It is a standard folklore result in computable analysis that can drastically simplify many arguments. It can be found in [51] for instance, but we include a proof for completeness. Proposition 2.1.1 Let X, Y be effective countably-based T 0 -spaces such that Y is effectively compact.

If R ⊆ X × Y is effectively closed, then its existential quantification

R ∃ := {x ∈ X : ∃y ∈ Y, (x, y) ∈ R} is effectively closed as well, If R ⊆ X × Y is effectively open, then its universal quantification R ∀ := {x ∈ X : ∀y ∈ Y, (x, y) ∈ R}
is effectively open as well.

Of course the two items are equivalent by taking complements, but we make both of them explicit as they are equally useful.

Proof. We prove the second item, the first one is obtained by taking complements. As R is effectively open, there exists a c.e. set E ⊆ N 2 such that R = (i,j)∈E B X i × B Y j . One has

x ∈ R ∀ ⇐⇒ {x} × Y ⊆ R (2.1) ⇐⇒ ∃ finite set L ⊆ E, {x} × Y ⊆ (i,j)∈L B X i × B Y j (2.2)
because {x} × Y is compact. Let F be the collection of (indices of) finite sets L ⊆ E such that Y ⊆ (i,j)∈L B Y j . As Y is effectively compact, F is a c.e. set. For each L ∈ F, let U L = (i,j)∈L B X i . The equivalence (2.2) implies that R ∀ = L∈F U L , which is an effective open set.

Descriptive set theory and its effective version provide notions of complexity for subsets of topological spaces ( [START_REF] Moschovakis | Descriptive Set Theory[END_REF]). We have already seen the classes Σ 0 1 and Π 0 1 in Definition 2.1.3, we will also need another class. Definition 2.1.4. A set A in an effective countably-based T 0 -space (X, τ,

(B i ) i∈N ) is Σ 0 2 if it can be expressed as A = n∈N A n \ B n
where A n , B n are uniformly effective closed sets (i.e. Π 0 1 ).

Observe that one can equivalently require A n , B n to be effectively open, as

A n \ B n = (X \ B n ) \ (X \ A n ).
This class is traditionally defined on computable metric spaces (those spaces are defined in Section 2.1.1 below), where it is equivalently defined without the sets B n . The definition given here was proposed for non-Hausdorff spaces by Scott [START_REF] Scott | Data types as lattices[END_REF] and Selivanov [START_REF] Victor | Towards a descriptive set theory for domain-like structures[END_REF], to make sure that the classes Σ 0 1 and Π 0 1 are contained in the class Σ 0 2 . Note that the non-effective classes are defined in Section 2.3.

Classes of Computable T 0 -spaces

Each topological separation axiom has a computable version, we recall some of them. These definitions can be found in [START_REF] Schröder | Effective metrization of regular spaces[END_REF] for instance. Definition 2.1.5.

1. A computable metric space is a tuple (X, d, α), where (X, d) is a metric space and α : N → X is a dense sequence in (X, d) such that the real numbers d(α i , α j ) are uniformly computable (i.e. the function (i, j) → d(s i , s j ) is computable). A computable Polish space is a computable metric space whose metric is complete.

2. An effective countably-based T 0 -space (X, τ,

(B i ) i∈N ) is computably Hausdorff if the diagonal ∆ = {(x, x) : x ∈ X} is an effectively closed subset of X × X, i.e. if there exists some c.e. set D ⊆ N 2 satisfying (X × X) \ ∆ = {(x, y) ∈ X × X : x = y} = (i,j)∈D B i × B j .
Recall that if (X, d) is a metric space, x ∈ X and r > 0, then we consider the open ball B(x, r) = {y ∈ X : d(x, y) < r} and the closed ball B(x, r) = {y ∈ X : d(x, y) ≤ r}. The following facts are standard and can be found in [51] for instance:

A computable metric space is an effective countably-based T 0 -space, by taking as basis the metric open balls B d (α i , q) with q > 0 a rational number; these balls are called the rational balls, A computable metric space is computably Hausdorff, In a computably Hausdorff space which is effectively compact, a set is semicomputable (i.e., effectively compact) if and only if it is effectively closed.

The next fact is another powerful feature of effective compactness: in certain situations, if a computable function is injective, then its inverse is automatically computable. Proposition 2.1.2

Let X be an effective countably-based T 0 -space, Y a computably Hausdorff space and K ⊆ X a semicomputable compact set.

If f : K → Y is a computable injective function, then its inverse f -1 : f (K) → K is computable.
Proof. It is a folklore result but we include a proof for completeness. We need to show that for a basic open set B i of X, its preimage under

f -1 is effectively open in f (K), uniformly in i. The preimage is precisely f (B i ∩ K). As f is injective, f (B i ∩ K) = f (K) \ f (K \ B i ). As K \ B i is semicomputable, its image f (K \ B i ) is semicomputable as well.
As Y is computably Hausdorff, f (K \ B i ) is effectively closed, so its complement is effectively open, as wanted.

The Hilbert Cube

The Hilbert cube will play a central role in the next chapters, because every computable metric space computably embeds into it, so one can work in this space without loss of generality. Definition 2.1.6. The Hilbert cube is the space Q = [0, 1] N endowed with the product topology, induced by the complete metric

d Q (x, y) = i 2 -i |x i -y i |
where x = (x 0 , x 1 , . . .) and y = (y 0 , y 1 , . . .). A point x ∈ Q is rational if its coordinates are rational and it has finitely many non-zero coordinates. The set of rational points is dense in Q.

Here are some important facts about the Hilbert cube. The Hilbert cube Q satisfies the following properties: 1. Let (α i ) i∈N be a computable enumeration of the points of Q having rational coordinates, finitely many of them being non-zero. It makes (Q, d Q , α) a computable metric space and even a computable Polish space, 2. The Hilbert cube is effectively compact, 3. Therefore, a set X ⊆ Q is semicomputable iff it is effectively closed, 4. Every computable metric space embeds effectively into the Hilbert cube. More precisely, for every computable metric space (X, d, α) there exists a computable embedding f :

X → Q such that f -1 is computable, defined as f (x) = (1/(1 + d(x, α i ))) i∈N ,
5. The points of Q can be multiplied by factors in [0, 1], and convex combinations can be performed in Q.

In particular, the fourth item implies that the semicomputable compact subsets of arbitrary computable metric spaces are no more general than the semicomputable compact subsets of Q.

The space of continuous functions from Q to itself Definition 2.1.7. Let C(Q) be the space of continuous functions from Q to itself endowed with the complete separable metric

d(f, g) = max x∈Q d Q (f (x), g(x)).
If functions f, g are defined on a compact subset X ⊆ Q only, then we also define

d X (f, g) = max x∈X d Q (f (x), g(x)).
To simplify the notation, we may write d(f, g) rather than d X (f, g).

Remark 2.1.2

We will often use the following inequalities:

d(f • h, g • h) ≤ d(f, g), (2.3)
d(f • g, id Q ) ≤ d(f, id Q ) + d(g, id Q ). (2.4)
The second inequality holds because d(f

• g, id Q ) ≤ d(f • g, g) + d(g, id Q ) ≤ d(f, id Q ) + d(g, id Q ).

Computable Type

The notion of computable type takes its origins in an article by Miller [START_REF] Miller | Effectiveness for embedded spheres and balls[END_REF], was studied by Iljazović et al. in [START_REF] Iljazović | Chainable and circularly chainable co-r.e. sets in computable metric spaces[END_REF][START_REF] Iljazović | spheres and cells in computable metric spaces[END_REF][START_REF] Iljazovic | Compact manifolds with computable boundaries[END_REF][START_REF] Burnik | Computability of 1-manifolds[END_REF][START_REF] Iljazović | Semicomputable manifolds in computable topological spaces[END_REF][START_REF] Čičković | Chainable and circularly chainable semicomputable sets in computable topological spaces[END_REF][START_REF] Horvat | Computability of pseudo-cubes[END_REF][START_REF] Iljazović | Computability of graphs[END_REF][START_REF] Čelar | Computability of products of chainable continua[END_REF][START_REF] Čelar | Computability of glued manifolds[END_REF] and by us in [START_REF] Djamel | Computability of finite simplicial complexes[END_REF]6,[START_REF] Djamel | Strong computable type. Computability[END_REF][START_REF] Djamel | The surjection property and computable type[END_REF]. The first formal definition as given in [START_REF] Iljazović | Semicomputable manifolds in computable topological spaces[END_REF].

Let us present the language needed to formulate the definition of computable type. 20

Background on Computable Topology

In this thesis, all the topological spaces are compact metrizable spaces, which we may implicitly assume. A copy of such a space X in some space Z is a subset of Z that is homeomorphic to X. We will often consider copies in Q.

We recall the definition of compact pairs, which is central in this thesis.

Definition 2.2.1. A compact pair (X, A) consists of a compact metrizable space X and a compact subset Note that the notion for pairs subsumes the notion for single sets, clearly by considering the pair (X, ∅).

A ⊆ X. A copy of a pair (X, A) in a topological space Z is a pair (Y, B) such that Y ⊆ Z is homeomorphic to X and

Example 2.2.1

The first example of a space which has computable type is n-spheres proved by Miller in [START_REF] Miller | Effectiveness for embedded spheres and balls[END_REF], more examples are given in Section 2.2.1 below.

Originally, two notions of computable type were studied in [START_REF] Iljazović | Chainable and circularly chainable co-r.e. sets in computable metric spaces[END_REF] and [START_REF] Čičković | Chainable and circularly chainable semicomputable sets in computable topological spaces[END_REF] respectively, using other spaces than the Hilbert cube. Precisely, A compact pair (X, A) has computable type on computable metric spaces if for every copy (Y, B) of the pair in any computable metric space Z, if (Y, B) is semicomputable then Y is computable, A compact pair (X, A) has computable type on computably Hausdorff spaces if for every copy (Y, B) of the pair in any computably Hausdorff space

Z, if (Y, B) is semicomputable then Y is computable. Remark 2.2.1
It is important to note that our approach to study computable type focuses exclusively on compact spaces, in contrast to certain existing results in the literature that establish a computable type property for non-compact spaces (e.g., [START_REF] Burnik | Computability of 1-manifolds[END_REF] or [START_REF] Iljazović | Computability of graphs[END_REF]). The assumption of compactness is pivotal, and extending the theory beyond compact spaces remains uncertain. Previous studies have introduced slight variations in the notion of computable type, such as employing effective closedness rather than effective compactness, as seen in [START_REF] Iljazović | spheres and cells in computable metric spaces[END_REF]. However, these variations necessitate additional assumptions on the ambient space, such as effective local compactness. In our thesis, we utilize the notion of effective compactness, which exhibits smoother behavior.

State of the Art

We summarize the main results in the literature in the following fact. The following spaces/pairs have computable type:

1. Spheres, balls with their bounding spheres [START_REF] Miller | Effectiveness for embedded spheres and balls[END_REF] and compact manifolds with or without boundary [START_REF] Iljazović | spheres and cells in computable metric spaces[END_REF][START_REF] Iljazovic | Compact manifolds with computable boundaries[END_REF][START_REF] Burnik | Computability of 1-manifolds[END_REF][START_REF] Iljazović | Semicomputable manifolds in computable topological spaces[END_REF][START_REF] Čelar | Computability of glued manifolds[END_REF],

2. Graphs with their endpoints [START_REF] Iljazović | Computability of graphs[END_REF],

3. Chainable continua between two points [START_REF] Iljazović | Chainable and circularly chainable co-r.e. sets in computable metric spaces[END_REF][START_REF] Čičković | Chainable and circularly chainable semicomputable sets in computable topological spaces[END_REF],

4. Circularly chainable continua which are not chainable [START_REF] Iljazović | Chainable and circularly chainable co-r.e. sets in computable metric spaces[END_REF][START_REF] Čičković | Chainable and circularly chainable semicomputable sets in computable topological spaces[END_REF],

5. Pseudo n-cubes [START_REF] Horvat | Computability of pseudo-cubes[END_REF],

6. Products of chainable continua [START_REF] Čelar | Computability of products of chainable continua[END_REF].

Vietoris Topology, Topological Invariant and Descriptive Complexity

In this discussion, we will explore an intriguing area of mathematics that involves the study of topological spaces and their relationship to topological invariants. Specifically, we will be interested in the concept of the Descriptive Set Theory (DST) of topological invariants, where we consider placing topologies on the hyperspace of the Hilbert cube and examine compact metric spaces as elements of this space. Now, let us delve into the detailed definitions.

Vietoris Topology Definition 2.3.1. Let K(Q) be the hyperspace of Q, i.e. the space of non-empty compact subsets of Q. It can be equipped with:

1. The upper Vietoris topology τ upV generated by the sets of the form

{K ∈ K(Q) : K ⊆ U },
where U ranges over the open subsets of Q, 2. The lower Vietoris topology τ lowV generated by the sets of the form

{K ∈ K(Q) : K ∩ U = ∅},
where U ranges over the open subsets of Q, 3. The Vietoris topology τ V generated by τ upV and τ lowV .

Note that the upper and lower Vietoris topologies are incomparable. These three topological spaces are effective countably-based T 0 -spaces. A countable basis (B

τ upV i
) i∈N of τ upV is obtained by taking U among the finite unions of rational balls of Q. A countable sub-basis (B τ lowV i ) i∈N of τ lowV is obtained by taking U among the rational balls of Q, and adding K(Q) to the subbasis. A countable basis for the Vietoris topology is obtained by taking finite intersections of these sets (the basis of τ upV and the sub-basis of τ lowV ). Using the basis (B i ) i∈N 22 2. Background on Computable Topology of Q, the elements of the sub-basis of (K(Q), τ V ) are of the form

   K : K ⊆ k∈{1,...,n} B i k , K ∩ B j = ∅   
for some i k s, j and n.

The Vietoris topology is also induced by the Hausdorff distance between non-empty compact sets in the Hilbert cube defined as follows.

Definition 2.3.2. Let A, B ⊆ Q be two non-empty compact sets, the Hausdorff distance between A and B is defined by:

d H (A, B) = max max a∈A min b∈B d Q (a, b), max b∈B min a∈A d Q (a, b) .
The space (K(Q), τ V ) is a computable Polish space, by taking the Hausdorff metric, and the dense sequence of finite sets of rational points of Q. Moreover, these three spaces are effectively compact (see Section 5 in [START_REF] Iljazović | Computability of subsets of metric spaces[END_REF] for instance; it is sufficient to prove it for the stronger topology τ V ).

Each computability notion of compact set can then be seen as the notion of computable point in one of these topologies. In particular, for a compact set

K ⊆ Q, K is semicomputable ⇐⇒ K is a computable element of (K(Q), τ upV ), K is computably ouvert ⇐⇒ K is a computable element of (K(Q), τ lowV ), K is computable ⇐⇒ K is a computable element of (K(Q), τ V ).
It is easy to get confused with the many different uses of the word "computable", but we will always make it clear what meaning is used.

We use the Vietoris topology rather than the Hausdorff metric because when measuring the descriptive complexity of some property (see Section 2.3), the Vietoris open sets appear more naturally (for instance, being disconnected is about open sets that cover and intersect the set).

Topological Invariant and Property of Pairs Definition 2.3.3. A topological invariant, or shortly an invariant, is a property

P ⊆ K(Q) such that for X, Y ∈ K(Q), if X ∈ P and Y is homeomorphic to X, then Y ∈ P.
This easily extends to pairs as follows.

Definition 2.3.4. A property of compact pairs is a subset P of K(Q) × K(Q). It is a topological invariant, if for every (X, A) ∈ P, every copy (Y, B) of (X, A) is in P. Example 2.3.1
The set of copies of a space or a pair is a topological invariant.

Descriptive Complexity

Let us briefly clarify how the descriptive complexity of topological invariants is measured.

Every compact metrizable space embeds in the Hilbert cube Q = [0, 1] N . Therefore, such a space can be described as a compact subset of Q.

We fix a topology on K(Q), which is either the Vietoris or the upper Vietoris topology. The descriptive complexity of a property of compact sets P ⊆ K(Q) in that topology is defined as follows.

It is Σ 0 1 if it is open, it is Π 0 1 if it is closed, it is Σ 0 2 if it
is a countable union of differences of Π 0 1 sets (equivalently, a countable uninon of Π 0 1 sets when K(Q) is endowed with the Vietoris topology). Effective complexity classes Π 0 1 and Σ 0 2 can be defined by requiring that the corresponding description of the set can be produced by a program (see Definitions 2.1.3 and 2.1.4 above). Specifically, P ⊆ K(Q) is Σ 0 1 if there exists a program enumerating an infinite list of basic open sets (encoded in the obvious way) whose union is

P; it is Π 0 1 if its complement is Σ 0 1 . A property P is Σ 0 2
in the Vietoris topology if P = n P n where each P n is Π 0 1 , uniformly in n (i.e. there is a single program describing P n on input n); it is Σ 0 2 in the upper Vietoris topology if it is a countable union of differences which can be enumerated by a program of Π 0 1 properties . We use the same notation for complexity classes in the Vietoris and upper Vietoris topologies and in the sequel we make it explicit which topology is used.

Example 2.3.2

It is not difficult to see that being disconnected a Σ 0 2 invariant in the upper Vietoris topology.

State of the Art

Let us mention previous works on the descriptive complexity of topological invariants. Ajtaj and Becker proved that path-connectedness is Π 1 2 -complete. Becker proved that for compact subsets of R 2 , simply-connectedness is Π 1 1 -complete ( [START_REF] Becker | Descriptive set theoretic phenomena in analysis and topology[END_REF][START_REF] Kechris | Classical Descriptive Set Theory[END_REF]). Debs and Saint-Raymond proved that connectedness is Π 0 1 , local connectedness if Π 0 3 , and gave another, independent proof that pathconnectedness is Π 1 2 [START_REF] Debs | The descriptive complexity of connectedness in Polish spaces[END_REF]. Lupini, Melnikov and Nies [START_REF] Lupini | Computable topological abelian groups[END_REF] and Downey, Melnikov [START_REF] Downey | Effectively compact spaces[END_REF] showed that the Čech cohomology groups are computable; in particular, whether the nth Čech cohomology group Ȟn (X) is non-trivial is Σ 0 2 . The general theory of the descriptive complexity of invariants was studied by Vaught [START_REF] Vaught | Invariant sets in topology and logic[END_REF].

When more information on the space is available, for instance if the space is given by a finite triangulation, many topological invariants become decidable (however, note that the fundamental group of a simplicial complex is computably presentable, but its non-triviality is undecidable), and the appropriate notion of complexity is their computational complexity, typically their time complexity. For instance, the complexity of combinatorial invariants associated to 3-manifolds represented by triangulations was studied in [START_REF] Burton | Algorithms and complexity for turaev-viro invariants[END_REF]. For simplicial complexes, the probem of deciding whether a homology group is non-trivial was proved to be NP-hard in [1]. The complexity of certain invariants associated to simply-connected spaces represented in an algebraic way was studied in [3].

Weihrauch reducibility

The goal of this section is to recall the definition of (strong) Weihrauch reducibility, to that end, we need some definitions (see [START_REF] Brattka | Weihrauch complexity in computable analysis[END_REF]). Definition 2.4.1. Let f, g :⊆ X ⇒ Y be multi-valued functions. We say that f is a strengthening of g and we write f g if dom(g) ⊆ dom(f ) and for every x ∈ dom(g), f (x) ⊆ g(x). A represented space (X, δ) is a set X together with a surjective partial function δ :

⊆ N N → X. A problem is a partial multi-valued function f :⊆ X ⇒ Y on represented spaces X and Y . Given represented spaces (X, δ X ), (Y, δ Y ), a problem f :⊆ X ⇒ Y and a function F :⊆ N N → N N . We say that F is a realizer of f and we write F f if δ Y F f δ X .
Now, we give the definition of (strong) Weihrauch reducibility. Definition 2.4.2. (Weihrauch reducibility) Let f and g be problems.

f is Weihrauch reducible to g, denoted f ≤ W g, if there exist computable func- tions K :⊆ N N → N N and H :⊆ N N × N N → N N such that for every G g, H • (id, G • K) f, where (id, G • K) : N N → N N × N N is defined as (id, G • K)(p) = (p, G(K(p))). f is strongly Weihrauch reducible to g, denoted f ≤ sW g, if there exist computable functions H, K :⊆ N N → N N such that for every G g, H • G • K f .
Clearly strong Weihrauch reducibility implies Weihrauch reducibility. Note that ≤ W and ≤ sW are preorders, i.e. they are reflexive and transitive. The corresponding equivalences are denoted by ≡ W and ≡ sW respectively and the symbols < W and < sW are used for strict reducibilities, respectively.

Strong Weihrauch reducibility relative to some oracle is usually denoted by ≤ t sW (where t stands for topological ).

Example 2.4.1

Let 1 denotes the Weihrauch degree of the identity id : [START_REF] Brattka | Weihrauch complexity in computable analysis[END_REF]).

N N → N N . If a problem f is computable then f ≤ W 1 (it is in fact an equivalence, see

Chapter 3

Background on General and Algebraic Topology

I

t turns out that the notion of computable type is very related to some notions in algebraic topology, this firstly appeared in the work of Miller (see [START_REF] Miller | Effectiveness for embedded spheres and balls[END_REF]), where he proved that n-spheres have computable type using classical results about homology of the complement of n-spheres, and Iljazović's utilization of some form of Brouwer's fixed-point theorem in [START_REF] Iljazović | spheres and cells in computable metric spaces[END_REF][START_REF] Iljazovic | Compact manifolds with computable boundaries[END_REF]. Many of our results use different notions in algebraic topology such as homotopy and homology, that is why we need to provide a brief background about algebraic topology.

This chapter provides an overview of the foundational concepts and results in general and algebraic topology that we use, drawing upon influential textbooks in the field. "Algebraic Topology" by Allen Hatcher [START_REF] Hatcher | Algebraic Topology[END_REF] offers a comprehensive introduction, covering homotopy theory, fundamental groups, and homology and cohomology theories. James Munkres' "Topology" provides a solid foundation in general topology, which is essential for studying algebraic topology [START_REF] Munkres | Topology. Featured Titles for Topology[END_REF]. "Topology and Geometry" by Glen E. Bredon [START_REF] Bredon | Topology and Geometry[END_REF] offers a comprehensive resource on algebraic topology, encompassing topics such as homotopy theory and cohomology. These textbooks serve as valuable references, equipping readers with the necessary tools to comprehend the subsequent analyses and investigations presented in the thesis.

Point-Set Topology

In addition to some basics in point-set topology in Section 3.1, we include in this preliminary statements and proofs of results that are classical or folklore, but which are stated slightly differently in the literature or for which we found no reference, and therefore deserve some explanations. Section 3.2 is about the cone and join of spaces, Section 3.3 contains results about homotopy of functions Section 3.4 contains Hopf's extension and classification theorems. Finally, in Section 3.5 we recall the notion of simplicial complexes.

Point-Set Topology

In this section we recall some classical definitions in point-set topology.

Notation. Here are some notations.

Let (X, τ ) be a topological space and A ⊆ X be a subset. Let cl τ (A) be the closure of A in (X, τ ) and int X (A) the interior of A in X.

If two topological spaces X and Y are homeomorphic, we write

X ∼ = Y . If (X, d) is a metric space, A ⊆ X and r > 0, let N (A, r) = N r (A) = {x ∈ X : d(x, A) < r}.

Connectedness and Dimension

Definition 3.1.1. In a topological space X, a subset A is connected if there is no disjoint open sets U, V ⊆ X both intersecting A and such that A ⊆ U ∪V . A connected component of X is a maximal connected subset.
We say that it is trivial if it is a singleton. A topological space X has dimension at most n if for every open cover (U i ) i∈N of X, there exists an open cover (V i ) i∈N of X such that each V i is contained in some U j and at most n + 1 sets V i 's have non-empty intersection.

Example 3.1.1

The n-dimensional sphere has dimension n.

Pointed Space, Wedge Sum and Disjoint Union Definition 3.1.2. A pointed topological space is a pair (X, x 0 ) where X is a topological space and x 0 ∈ X is a distinguished point. If (X, x 0 ) and (Y, y 0 ) are pointed spaces, then their wedge sum is the space X ∨ Y obtained by attaching X and Y at their distinguished points, which are identified. If X, Y are topological spaces, then their disjoint union is X Y . The disjoint union topology on X Y is defined as the finest topology on X Y for which the canonical injections X → X Y and Y → X Y are continuous.

Note that in the case where X and Y are embedded in Q, to build disjoint copies of X and Y for instance, one can take {0} × X and {1} × Y (which are still contained in

[0, 1] × Q ∼ = Q). Example 3.1.2
The Wedge sum of two segments which have their distinguished points in their endpoints is again a segment.

(Proper) Subpair and Function between Pairs

If (X, A) and (Y, B) are two pairs embedded in a topological space Z, then we say that (X, A) is a subpair of (Y, B), or is contained in (Y, B) if X ⊆ Y and A ⊆ B. We then write (X, A) ⊆ (Y, B). We say that (X, A) is strictly contained in (Y, B), or is a proper subpair of (Y, B), if in addition X = Y (note that A may equal B), and we write (X, A) (Y, B).

(X, A) → (Y, B) is a function f : X → Y such that f (A) ⊆ B. Example 3.1.3 Let (X, A) be a pair, every function f : X → X which is the identity on A is a function of pairs f : (X, A) → (X, A).
Quotient Space Definition 3.1.4. Let X be a topological space, and let ∼ be an equivalence relation on X. The quotient X/ ∼ is the set of equivalence classes of elements of X.

The equivalence class of x ∈ X is denoted [x]. The canonical quotient map is the surjection q : x ∈ X → [x]. The quotient space is X/ ∼ equipped with the quotient topology, whose open sets are the subsets O ⊆ X/ ∼ such that q -1 (O) is open in X. Let A be a subspace of X. The quotient space of X by A, denoted X/A, is X/ ∼ A where for every x, y ∈ X, x ∼ A y iff x = y or x, y ∈ A. Example 3.1.4
The quotient of the n-ball by its bounding (n -1)-sphere is the n-sphere.

Manifolds

We first recall the definition of a manifold.

Cone and join

The cone of a space and the join of two spaces are classical constructs in topology. We present their counterparts for pairs. Definition 3.2.1. Let X be a topological space, its cylinder is the space Cyl(X) = X ×[0, 1] endowed with the product topology.

Example 3.2.1

The cylinder of the line segment is the disk.

Cone and Cone Pair

Definition 3.2.2. (Cone) Let X be a topological space. The cone of X is the quotient of X × [0, 1] under the equivalence relation (x, 0) ∼ (x , 0). The open cone of X is the space OC(X) = C(X) \ X, where X is embedded in C(X) as X × {1}. The equivalence class of X × {0} is called the tip of the cone. If X is embedded in Q, then a realization of C(X) in Q ∼ = [0, 1] × Q is given by C(X) ∼ = {(t, tx) : t ∈ [0, 1], x ∈ X}. Definition 3.2.3. (Cone pair) Let (X, A) be a pair. Its cone pair is the pair C(X, A) = (C(X), X ∪ C(A)). Its open cone pair is the pair OC(X, A) = (OC(X), OC(A)).
In particular,

C(L, ∅) = (C(L), L) OC(L, ∅) = (OC(L), ∅).

Special Symbol

We introduce a symbol , which will be treated as a space but is purely formal. It enables to consider a singleton as a cone and is a unit for the join (see Section 3.2.2). It can be thought as the boundary of the singleton, as in the definition of reduced homology. It is analogous to the empty simplicial complex {∅}, which behaves differently from the void complex ∅.

Definition 3.2.4. We define C( ) = OC( ) = {0} (3.1)
where the point 0 is thought as the tip of a degenerate cone. We take the convention that S -1 = .

Example 3.2.2

Let n ∈ N. One has

C(S n , ∅) ∼ = (B n+1 , S n ), C(B n , S n-1 ) ∼ = (B n+1 , S n ),
with the tip at the center of B n+1 and in S n respectively. Note that the particular case n = 0 is consistent with the convention S -1 = .

Join

Definition 3.2.5. If X, Y are topological spaces, then their join X * Y is the quotient of X × Y × [0, 1] under the equivalence relation (x, y, 0) ∼ (x, y , 0) and (x, y, 1) ∼ (x , y, 1).

Example 3.2.3

The join of two disjoint points is an interval. The join of a point and an interval is a triangle.

Remark 3.2.1

In the literature, a convention is to let ∅ be a unit for the join, i.e. to define X * ∅ as X. Definition 3.2.5 rather makes ∅ an absorbing element, the unit element will be :

X * ∅ = ∅ * X = ∅, X * = * X = X.
If X and Y are embedded in Q, then X * Y can be realized as

X * Y ∼ = {(t, (1 -t)x, ty) : t ∈ [0, 1], x ∈ X, y ∈ Y }.
We also realize

X * ∼ = {(0, x, 0) : x ∈ X} and * Y ∼ = {(1, 0, y) : y ∈ Y }.
The cone and join constructs are intimately related: C(X) = X * {0}. It is proved in Brown [START_REF] Brown | Topology and Groupoids[END_REF] (Corollary 5.7.4, §5.7, p. 196) that a product of cones is a cone. More precisely, let X, Y be compact metrizable spaces. There is a homeomorphism

C(X * Y ) → C(X) × C(Y ) which restricts to a homeomorphism X * Y → (C(X) × Y ) ∪ (X × C(Y )).
We will use the following consequences: Proposition 3.2.1 Let X, Y be compact metrizable spaces, or . One has

C(X) × C(Y ) ∼ = C(X * Y ) OC(X) × OC(Y ) ∼ = OC(X * Y ).
Proof. The first equality is Corollary 5.7.4 in [START_REF] Brown | Topology and Groupoids[END_REF] for the general case. We need to check the particular cases when

Y is ∅ or . One has C(X) × C(∅) = ∅ = C(X * ∅) and C(X) × C( ) = C(X) × {0} = C(X * ),
and the symmetric cases are similar. The second equality can be proved as follows for X, Y = ,

OC(X) × OC(Y ) = (C(X) \ X) × (C(Y ) \ Y ) = (C(X) × C(Y )) \ ((C(X) × Y ) ∪ (X × C(Y ))) ∼ = C(X * Y ) \ (X * Y ) = OC(X * Y ), and when Y = , OC(X) × OC( ) ∼ = OC(X) = OC(X * ).

Join of pairs

The previous result extends to cone pairs. Note that in all the pairs (X, A) and (Y, B) that we consider, X and Y are never ∅ or . Definition 3.2.6. The join of two pairs (X, A) and (Y, B) is

(X, A) * (Y, B) = (X * Y, X * B ∪ A * Y ). Example 3.2.4
Let (I, p) be a segment and one of its endpoints, one has (I, p) * (I, p) = (I * I, I * p ∪ p * I) which is topologically a tetrahedron with its base.

The cone pair is a particular case of the join of pairs:

(X, A) * ({0}, ) = (C(X), X ∪ C(A)) = C(X, A), (X, ∅) * ({0}, ) = (C(X), X) = C(X, ∅). Proposition 3.2.2 A product of (open) cone pairs is a (open) cone pair: C(X, A) × C(Y, B) ∼ = C((X, A) * (Y, B)) OC(X, A) × OC(Y, B) ∼ = OC((X, A) * (Y, B)). Proof. The first component of C(X, A) × C(Y, B) is C(X) × C(Y ) = C(X * Y ) by Proposition 3.2.1. Its second component is (C(X) × (Y ∪ C(B))) ∪ ((X ∪ C(A)) × C(Y )) = (C(X) × Y ) ∪ (X × C(Y )) ∪ (C(X) × C(B)) ∪ (C(A) × C(Y )) ∼ = (X * Y ) ∪ C(X * B) ∪ C(A * Y ) ∼ = (X * Y ) ∪ C(X * B ∪ A * Y ). The first component of OC(X, A) × OC(Y, B) is OC(X) × OC(Y ) ∼ = OC(L) by Proposition 3.2.1. Its second component is (OC(X) × OC(B)) ∪ (OC(A) × OC(Y )) ∼ = OC(X * B) ∪ OC(A * Y ) ∼ = OC(X * B ∪ A * Y ).
3.3 Homotopy and Absolute Neighborhood Retracts (ANRs)

Homotopy

Definition 3.3.1. A homotopy between two continuous functions f, g : X → Y is a contin- uous function H : [0, 1] × X → Y such that H(0, x) = f (x) and H(1, x) = g(x) for all x ∈ X.
Two functions are homotopic if there exist a homotopy between them. A function is nullhomotopic if it is homotopic to a constant function.

A homotopy H : [0, 1] × X → Y can be seen as a family of continuous functions

h t : X → Y where t ∈ [0, 1] and (t, x) → h t (x) is continuous, by taking h t (x) = H(t, x). Example 3.3.1
The identity from a ball to itself is null-homotopic.

A function f : Y → Z is null-homotopic if and only if it can be extended to a continuous function f : C(Y ) → Z, where Y is embedded in C(Y ) as Y × {1}.
Homotopy between functions is an equivalence relation, if X, Y are topological spaces and f : X → Y is continuous, then we denote by [f ] the homotopy class of f . Let [X; Y ] be the set of homotopy classes of continuous functions from X to Y .

Homotopy extension property

Let us recall an important property of pairs. Definition 3.3.2. A pair (X, A) has the homotopy extension property if for every topological space Y , for every continuous function f : X → Y and every homotopy h

t : A → Y such that h 0 = f | A , there exists a homotopy H t : X → Y such that H 0 = f and H t | A = h t . Example 3.3.2
For every n ∈ N, the pair (B n+1 , S n ) has the homotopy extension property. Definition 3.3.3. Let (X, A) be a pair. A retraction r : X → A is a continuous function such that r| A = id A . If a retraction exists, then we say that A is a retract of X. Hatcher [30], Proposition A.18, p. 533). If (X, A) is a CW pair, then it has the homotopy extension property (see Hatcher [START_REF] Hatcher | Algebraic Topology[END_REF], Proposition 0.16, p. 15). Definition 3.3.4. Let X be a topological space and (Y, d) a metric space. For > 0, anhomotopy is a homotopy function h t : X → Y such that d(h t (x), h 0 (x)) < for all x, t.

It is well-known that a pair (X, A) has the homotopy extension property if and only if

X × [0, 1] retracts to (X × {0}) ∪ (A × [0, 1]) (see
The homotopy extension property actually implies a controlled version. Lemma 3.3.1 Let (X, A) be a pair satisfying the homotopy extension property.

If (Y, d) is a metric space, f : X → Y is continuous, h t : A → Y is an -homotopy such that h 0 = f | A , then there exists an -homotopy H t : X → Y such that H 0 = f and H t | A = h t .
A similar result with a similar proof holds for arbitrary compact pairs (X, A) when Y is an ANR (Theorem 4.1.3, p. 265 in [START_REF] Van Mill | The Infinite-Dimensional Topology of Function Spaces[END_REF]).

Proof. As (X, A) has the homotopy extension property, there exists a retraction r : X × [0, 1] → (X ×{0})∪(A×[0, 1]) and a homotopy H t : X → Y extending H 0 and h t . We define a retraction r :

X × [0, 1] → (X × {0}) ∪ (A × [0, 1]) so that H t (x) = H • r (x, t) is an -homotopy. Let U = {x ∈ X : ∀t ∈ [0, 1], d(H t (x), H 0 (x)) < }. As [0, 1] is compact, U is an open set. Moreover, U contains A. Let δ : X → [0, 1]
be a continuous function such that δ(x) = 1 for x ∈ A and δ(x) = 0 for x ∈ X \ U . We define r as follows:

r (x, t) = r(x, tδ(x)).
First, r is a retraction: r (x, 0) = r(x, 0) = (x, 0) and for x ∈ A, r (x, t) = r(x, t) = (x, t). Therefore, H is a homotopy extending H 0 and h t . We claim that H is an -homotopy, which means that d(H t (x),

H 0 (x)) < . If x / ∈ U , then H • r (x, t) = H • r(x, 0) = H(x, 0) = H 0 (x). If x ∈ U , then H • r (x, t) = H • r(x, tδ(x)) is -close to H 0 (x).

Absolute Neighborhood Retracts (ANRs)

We recall the definition of Absolute Neighborhood Retracts (ANRs).

This important notion was introduced by Borsuk [START_REF] Borsuk | Über eine Klasse von lokal zusammenhangende Räumen[END_REF] and plays an eminent role in algebraic topology. They have many interesting properties that we will use in our proofs. Moreover, it has very useful computability-theoretic consequences, which we will take advantage of.

Definition 3.3.5. Let X be a Hausdorff compact space. 1. X is an absolute retract (AR) if every copy of X in Q is a retract of Q, 2. X is an absolute neighborhood retract (ANR) if every (equivalently, some) copy X of X in Q is a retract of a neighborhood of X . Fact 3.3.1
We recall some classical facts (see [START_REF] Hanner | Some theorems on absolute neighborhood retracts[END_REF] and [START_REF] Van Mill | The Infinite-Dimensional Topology of Function Spaces[END_REF]).

1. The n-dimensional ball is an AR, 2. The n-dimensional sphere is an ANR, 3. If Y is an AR and (X, A) is a pair, then every continuous function

f : A → Y has a continuous extension F : X → Y , 4. If Y is an ANR and (X, A) is a pair, then every continuous function f : A → Y has a continuous extension F : U → Y where U ⊆ X is a neighborhood of A.
The topologist's sine curve, or the set {0} ∪ {1/n : n ∈ N} are examples of spaces that are not ANRs.

ANRs and Homotopy

ANRs interact nicely with the notion of homotopy.

If two functions to a compact ANR are close to each other, then they are homotopic (Theorem

4.1.1 in [60]). Lemma 3.3.2
Let Y ⊆ Q be a compact ANR. There exists α > 0 such that for every space X, if f, g : X → Y are continuous and d X (f, g) < α, then f, g are homotopic.

Whether a function to an ANR has a continuous extension only depends on the homotopy class of the function. This is Borsuk's homotopy extension theorem (Theorem 1.4.2 in [60] or Theorem V.3.1 in [START_REF] Borsuk | Theory of retracts[END_REF]). Let Y be a compact metric ANR. For every > 0 there exists δ > 0 such that for all pairs (X, A) and all continuous functions f, g :

A → Y with d A (f, g) < δ, if f has a continuous extension F : X → Y then g has a continuous extension G : X → Y such that d X (F, G) < .
The following result is Theorem 4.1.1 in [START_REF] Van Mill | The Infinite-Dimensional Topology of Function Spaces[END_REF].

Lemma 3.3.4
Let Y be a compact metric ANR. For every > 0 there exists δ > 0 such that for all spaces X, all continuous functions f, g : X → Y satisfying d X (f, g) < δ are -homotopic.

The next theorem is a direct consequence of the main result in [START_REF] Whitehead | Note on a theorem due to Borsuk[END_REF]. Theorem 3.3.2. If X and A ⊆ X are compact ANRs, then (X, A) has the homotopy extension property and X/A is an ANR.

Pair vs quotient

In many cases, a pair (X, A) can be equivalently replaced by the quotient space X/A. The following proposition is an instance of this fact, and is a combination of Proposition 4A.2 and Example 4A.3 in [START_REF] Hatcher | Algebraic Topology[END_REF].

Let (X, A) be a pair. There is a one-to-one correspondence between the continuous functions f : X → S n that are constant on A and the continuous functions g :

X/A → S n . If f : X → S n is constant on A then we denote by f : X/A → S n the function satisfying f = f •q, where q : X → X/A is the quotient map. Proposition 3.3.1 Let n ≥ 1 and let s be a distinguished point of S n . A function of pairs f : (X, A) → (S n , s) is null-homotopic if and only if f : X/A → S n is null-homotopic.
Proof. Let p be the equivalence class of A in X/A. Note that f can also be seen as a function of pairs f 2 : (X/A, p) → (S n , s), which we denote differently to avoid confusions. The null-homotopy of f is easily equivalent to the null-homotopy of f 2 , inducing a null-homotopy of f . We need to show that a null-homotopy of f (a function between sets) implies a null-homotopy of f 2 (a function between pairs).

Let h t : X/A → S n be a homotopy from h 0 = f to a constant function h 1 . We use a different argument for n = 1 and for n = 1.

Homology and Cohomology

Assume that n = 1. The circle S 1 can be seen as the additive group R/Z with s as the 0 element. Let g t : (X/A, p) → (S 1 , s) be defined by g t (x) = h t (x) -h t (p) + h 0 (p). One easily checks that g 0 = h 0 = f , g t (p) = h 0 (p) = s for all t and g 1 (x) = s for all x, so g t is a null-homotopy of f 2 .

Assume that n

= 1. Let Y = [0, 1] × (X/A) and B = ({0, 1} × X/A) ∪ ([0, 1] × {p}) ⊆ Y . The null-homotopy h t is a function h : Y → S n . Let g : B → S n be the continuous function defined by g(t, x) = f (x) if t = 0, s otherwise.
A null-homotopy of f 2 is a continuous extension G : Y → S n of g. In order to show that such an extension exists, it is sufficient to show that g is homotopic to the restriction of h to B, by applying Borsuk's homotopy extension theorem (Theorem 1.4.2, p. 38 in [START_REF] Van Mill | The Infinite-Dimensional Topology of Function Spaces[END_REF]) and the fact that S n is an ANR.

So let us define a homotopy k t : B → S n from k 0 = g to k 1 = h| B . We decompose B as B 0 ∪ B 1 , where B 0 = {0, 1} × X/A and B 1 = [0, 1] × {p}. On B 0 , g and h coincide, and we define k t (z) = g(z) = h(z) for all t and all z ∈ B 0 . On B 1 , g has constant value s and the restriction of h to B 1 is a loop from s to itself. As S n is simply connected, that loop is contractible, i.e. there exists a homotopy from g| B 1 to h| B 1 , and we define k t | B 1 as this homotopy.

The homotopy

k t : B → S n is well-defined because it is consistent on B 0 ∩ B 1 = {0, 1} × {p}, where k t (0, p) = k t (1, p) = s.
Therefore, k t is continuous and is a homotopy from g to h| B . It can be extended to a homotopy from some extension of g to h, because S n is an ANR. The extension of g is a null-homotopy of f 2 .

Homology and Cohomology

We do not define homology and cohomology groups, but just recall the notations and some classical results. We refer to standard textbooks on algebraic topology for complete expositions of the concepts [START_REF] Hurewicz | Dimension Theory (PMS-4)[END_REF][START_REF] Bredon | Topology and Geometry[END_REF][START_REF] Hatcher | Algebraic Topology[END_REF]. This section is not essential and may be skipped by the reader who is unfamiliar with algebraic topology.

If X is a topological space, G an abelian group and n ∈ N, one can define the homology groups H n (X; G) and the cohomology groups H n (X; G), which are abelian groups. They are topological invariants that, informally, detect the n-dimensional holes of the space X.

Example 3.4.1

When X is the m-dimensional sphere S m , one has

H n (S m ; G) ∼ = H n (S m ; G) ∼ = G if n = m or n = 0, 0 otherwise,
where 0 is the trivial group with one element.

These groups are also called the singular homology and cohomology groups, to distinguish them from the groups from other homology and cohomology theories, such as the Čech homology and cohomology groups, denoted by Ȟn (X; G) and Ȟn (X; G) respectively. However, these different theories are equivalent for ANRs (Theorem 1 in [START_REF] Mardešić | Equivalence of singular and čech homology for anr-s. application to unicoherence[END_REF]). Theorem 3.4.1. If X is an ANR, then for all G and n,

H n (X; G) ∼ = Ȟn (X; G), H n (X; G) ∼ = Ȟn (X; G).
The result applies to compact manifolds with and without boundary, which are ANRs (Corollary A.9 in [START_REF] Hatcher | Algebraic Topology[END_REF]).

Hopf 's extension and classification theorems

We recall two classical results: Hopf's extension and classification theorem. They are usually expressed in terms of cohomology. The following statements using homology can be found in Hurewicz-Wallman [START_REF] Hurewicz | Dimension Theory (PMS-4)[END_REF]. They are stated in terms of Čech homology for compact spaces in [START_REF] Hurewicz | Dimension Theory (PMS-4)[END_REF]. We state them for compact ANRs using singular homology, which is equivalent to Čech homology for these spaces.

Hopf's extension theorem is Theorem VIII.1' in [START_REF] Hurewicz | Dimension Theory (PMS-4)[END_REF] ( §VIII.6, p. 147). Let T = R/Z be the circle group.

Theorem 3.4.2. Let n ∈ N. Let (X, A) be a compact pair, where X and A are ANRs and dim(X) ≤ n + 1. For a continuous function f : A → S n , the following statements are equivalent:

f has a continuous extension F : X → S n , ker i * ⊆ ker f * ,
where i : A → X is the inclusion map and

i * : H n (A; T) → H n (X; T) f * : H n (A; T) → H n (S n ; T)
are the homomorphisms induced by i and f . The equivalence holds for n = 0 or 1, without any dimension assumption about X.

For n = 0, Theorem 3.4.2 also holds when considering homomorphisms between reduced homology groups H 0 (•; T).

We present the argument for the particular case n = 0 or 1, which is not stated in [START_REF] Hurewicz | Dimension Theory (PMS-4)[END_REF] but follows from the proof.

Proof. Assume that n ≤ 1. We only need to prove the result for CW-pairs (X, A), because a pair of compact ANRs is homotopy equivalent to a CW-pair. We assume that ker i * ⊆ ker f * and prove that f has an extension. From Hopf's extension theorem for spaces of dimension at most n + 1, f has an extension to the n + 1-skeleton of X. For k ≥ n + 1, every function from S k to S n is null-homotopic (because n = 0 or 1), so f can be inductively extended to every cell of X. Theorem 3.4.3. Let n ∈ N. Let (X, A) be a compact pair, where X and A are ANRs and dim(X) ≤ n. For a continuous function f : (X, A) → (S n , s), the following statements are equivalent:

f is null-homotopic (as a function of pairs),

f * : H n (X, A; T) → H n (S n , s; T) is trivial.
The equivalence holds for n ≤ 1, without any dimension assumption about X.

When n = 0 and A = ∅, and only in that case, one indeed needs to consider the homomorphism between reduced homology groups. Proof. This result is usually stated in terms of cohomology, and for single spaces rather than pairs (for instance, Theorem VIII.2, §VIII.6, p. 149 in [START_REF] Hurewicz | Dimension Theory (PMS-4)[END_REF]). It can be derived from Theorem 3.4.2 as follows. Let g : X ∪ C(A) → S n be defined as g = f on X and g = s on C(A). A null-homotopy of f is a function G : C(X) → S n extending g. The pair (C(X), X ∪ C(A)) satisfies all the conditions of Hopf's extension theorem, so g has extension if and only if ker i * ⊆ ker g * . As C(X) is contractible, its reduced homology groups are trivial so ker i * is the whole group H n (X ∪C(A); T); therefore, g has an extension iff g * : H n (X ∪C(A); T) → H n (S n ; T) is trivial. Up to natural isomorphisms, f * is g * .

Simplicial Complexes

Definition 3.5.1. Let V = {0, . . . , n} and P + (V ) be the set of non-empty subsets of V . An abstract finite simplicial complex is a set S ⊆ P + (V ) such that if σ ∈ S and ∅ = σ ⊂ σ, then σ ∈ S. Its elements σ ∈ S are called the simplices of S. If σ ∈ S has n + 1 elements, then σ is an n-simplex. The vertices of S are the singletons {i} ∈ S. σ ∈ S is free if there exists exactly one σ ∈ S with σ σ . A subcomplex of S is an abstract simplicial complex contained in S. The support of a vector x = (x 0 , . . . , x n ) ∈ [0, 1] n+1 is supp(x) = {i : x i = 0} . The standard realization of an abstract simplicial complex S is the set

|S| = x = (x 0 , . . . , x n ) ∈ [0, 1] n+1 : i x i = 1, supp(x) ∈ S .
We say that a simplex σ in S is maximal if it is not contained in a higher-dimensional simplex of S. Definition 3.5.2. Any space homeomorphic to the standard realization of an abstract finite simplicial complex is called a finite simplicial complex. We often identify an abstract simplicial complex and its standard realization. A finite simplicial pair (X, A) consists of a finite simplicial complex X and a subcomplex A.

Example 3.5.1

For every n ∈ N, (B n+1 , S n ) is a finite simplicial pair.

In a finite simplicial complex, each vertex has a neighborhood which is usually called a star and is topologically a cone. Indeed, let (X, A) be the standard realization of a finite simplicial pair and v i = (0, . . . , 0, 1, 0, . . . , 0) be a vertex. A topological cone which is a neighborhood of v i can be of the form (K i , M i ) such that:

K i = {x ∈ X : x i ≥ 1/2}, M i = {x ∈ X : x i = 1/2} ∪ (K i ∩ A),
note that the coefficient 1/2 is arbitrary and could be replaced by any number in (0, 1). K is a topological cone: let L i = {x ∈ X :

x i = 1/2}, K i is a copy of the cone of L i , obtained from L i ×[0, 1]
by identifying all the points (l, 0) together. The point obtained by this identification is the tip of the cone and corresponds to the vertex v i . If N i = {x ∈ A : x i = 1/2}, then M i is the union of L i and of the cone of N i .

In the language of simplicial complexes, K i corresponds to the star of v i and L i to the link of v i . K i is homeomorphic to the union of simplices containing v i . Each such simplex has a face that does not contain v i , and L i is the union of these faces. The results stated in this chapter can be found in [START_REF] Djamel | Strong computable type. Computability[END_REF]6].

Part II

Main contributions

Introduction

T he objective of this chapter is to establish a structural understanding of the notion of computable type (and not to present new examples), with several goals in mind. Given the technical and challenging nature of establishing whether a space has computable type, there is a need for unified and simplified arguments for previous results, as well as general tools that can be applied to establish new results more effortlessly.

Our overarching aim is to emphasize the interplay between topology and computability that underlies the concept of computable type. The definition of computable type combines ideas from both topology and computability theory, and the arguments employed draw inspiration from these 4.2 Computable Type and the Hilbert Cube two fields. It would be illuminating to break down the arguments into two parts: a purely topological component and a computability-theoretic component.

Since computable type is inherently a topological invariant, it is crucial to explore its precise relationship with other topological invariants. A clearer connection with topology would allow for leveraging the extensive field of topology, particularly algebraic topology, in the investigation of computability-theoretic problems. Notably, the main results concerning computable type rely on classical topological results, such as Miller's utilization of the homology of the complement of the sphere in [START_REF] Miller | Effectiveness for embedded spheres and balls[END_REF], and Iljazović's utilization of some form of Brouwer's fixed-point theorem in [START_REF] Iljazović | spheres and cells in computable metric spaces[END_REF][START_REF] Iljazovic | Compact manifolds with computable boundaries[END_REF].

A deeper theoretical understanding of computable type has far-reaching consequences. It can provide additional insights into previous results, facilitate the derivation of new results with minimal effort, and shed light on the role of specific assumptions in the outcomes.

The chapter is organized as follows. In Section 4.2 we prove that some of the variations of computable type introduced in the literature are actually equivalent. In Section 4.3 we introduce the notion of strong computable type and develop the theory. We obtain characterizations using descriptive set theory on the hyperspace of compact subsets of the Hilbert cube. In Section 4.4 we exploit these characterizations and obtain several results that improve our understanding of strong computable type. In Section 4.5, we finish by studying how some spaces fail to have computable type.

Computable Type and the Hilbert Cube

In this section, we establish that the distinction between computable type on computable metric spaces and computably Hausdorff spaces is unnecessary, as these two notions are actually equivalent. Furthermore, we demonstrate that it suffices to consider the Hilbert cube alone, as defined in Definition 2.2.2. As a result, there is no longer a need to differentiate between these definitions, and the results pertaining to computable type on computable metric spaces seamlessly extend to computably Hausdorff spaces. For instance, the findings in [START_REF] Iljazović | Chainable and circularly chainable co-r.e. sets in computable metric spaces[END_REF][START_REF] Iljazovic | Compact manifolds with computable boundaries[END_REF] readily imply the results presented in [START_REF] Čičković | Chainable and circularly chainable semicomputable sets in computable topological spaces[END_REF][START_REF] Iljazović | Semicomputable manifolds in computable topological spaces[END_REF]). Theorem 4.2.1. For a compact pair (X, A), the following statements are equivalent:

1. (X, A) has computable type (on the Hilbert cube), 2. (X, A) has computable type on computable metric spaces,

(X, A) has computable type on computably Hausdorff spaces.

We now proceed with the proof of Theorem 4.2.1. The implications 3. ⇒ 2. ⇒ 1. are straightforward, we only need to prove 1. ⇒ 3. The idea is that a compact subspace of a Hausdorff space is metrizable, so embedding a compact set in a Hausdorff space implicitly induces an embedding in a metric space. We need to make this argument effective. We will use Schröder's effective version of Urysohn metrization theorem [START_REF] Schröder | Effective metrization of regular spaces[END_REF], so we need to introduce the computable version of regular spaces.

F k ) k∈N such that U k ⊆ F k ⊆ B n and B n = k U k .
Let (X, τ ) be an effective countably-based T 0 -space. If (X, τ ) is effectively compact and computably Hausdorff, then it is computably regular.

Proof. Let D ⊆ N 2 witness that the space is computably Hausdorff. Let n ∈ N. The set K = X \ B n is effectively compact, so one can compute an enumeration (L k ) k∈N of all the finite sets L ⊆ D such that K ⊆ (i,j)∈L B j . To each such L k associate the pair (U k , F k ) defined by

U k = (i,j)∈L k B i F k = X \ (i,j)∈L k B j .
By construction, U k and F k are uniformly effectively open and closed respectively.

Next,

U k is contained in F k because B i is disjoint from B j for all (i, j) ∈ L k ⊆ D. As X \ B n ⊆ (i,j)∈L k B j , one has F k = X \ (i,j)∈L k B j ⊆ B n . Finally, we show that B n = k U k . Let x ∈ B n . As x / ∈ K, the compact set {x}×K is contained in (i,j)∈D B i × B j ,
so there exists a finite set L ⊆ D such that {x} × K ⊆ (i,j)∈L B i × B j . This inclusion is still satisfied if one removes from L the pairs (i, j) such that x / ∈ B i . Let L ⊆ L be the result of this operation. There exists k such that L = L k , and x ∈ U k .

Schröder proved the following effective Urysohn metrization theorem (Theorem 6.1 in [START_REF] Schröder | Effective metrization of regular spaces[END_REF]): every computably regular space X admits a computable metric, i.e. a computable function d : X ×X → R which is a metric that induces the topology of the space (note that it does not mean that X is a computable metric space, because it may not contain a dense computable sequence). The proof of that result implies the next lemma. If (X, τ ) is a computably regular space, then there exists a computable injection h : X → Q.

Proof. The proof of Theorem 6.1 in [START_REF] Schröder | Effective metrization of regular spaces[END_REF] consists in building a computable sequence of functions g i : X → [0, 1] that separates points, i.e. such that for all x, y ∈ X with x = y, g i (x) = g i (y) for some i ∈ N. The metric d is then defined as d(x, y) = i 2 -i |g i (x) -g i (y)|. In our case, we simply define h(x) = (g i (x)) i∈N .

Corollary 4.2.1. If K is a semicomputable compact set in a computably Hausdorff space (X, τ ), then there exists a computable embedding h : K → Q. Its inverse is necessarily computable by Proposition 2.1.2.

Proof. Consider the subspace (K, τ K ) with the induced topology τ K = {U ∩ K : U ∈ τ }. Because it is a subspace of a computably Hausdorff space, it is computably Hausdorff as well. As K is semicomputable, as a set, it is effectively compact as a space. Therefore, Proposition 4.2.1 implies that (K, τ K ) is computably regular. By Lemma 4.2.1, there exists a computable injection h : K → Q. Now, we prove our theorem. Proof of Theorem 4.2.1. We prove 1. ⇒ 3. If (K, L) is a compact pair that has computable type (on the Hilbert cube), (X, τ ) is a computably Hausdorff space and f : K → X is an embedding such that f (K) and f (L) are semicomputable compact sets, then by Corollary 4.2.1, there exists a computable embedding h :

f (K) → Q such that h -1 is computable. Hence, h • f : K → Q is an embedding such that h • f (K) and h • f (L) are semicomputable compact sets. As (K, L) has computable type, the compact set h • f (K) is computable. The compact set f (K) is the image of the computable compact set h • f (K) by the computable function h -1 , so f (K) is a computable compact set.
As a result, (K, L) has computable type on computably Hausdorff spaces.

Strong Computable Type

The notion of computable type encounters a significant limitation: if a compact metrizable space does not possess a semicomputable copy within the Hilbert cube, it automatically has computable type without any substantial justification. Since there are only countably many semicomputable sets, the majority of compact metrizable spaces have computable type in a rather trivial manner. Consequently, the prospect of deriving meaningful characterizations for this notion becomes dim.

To overcome this issue, we introduce a more robust and powerful concept: computable type relative to any oracle. This refined definition resolves the aforementioned drawback and offers the advantage of facilitating topological analysis, particularly concerning the topologies applied to the hyperspace K(Q) comprising compact subsets of Q. These topological considerations enable us to attain valuable characterizations from a topological perspective.

Definition

We first define the notion of strong computable type, which is a simple relativization of Definition 2.

(Y, B) of (X, A) in Q, if (Y, B) is semicomputable relative to O, then Y is computable relative to O. Remark 4.3.1
All the spaces which were proved to have computable type in the literature [START_REF] Miller | Effectiveness for embedded spheres and balls[END_REF][START_REF] Iljazović | Chainable and circularly chainable co-r.e. sets in computable metric spaces[END_REF][START_REF] Iljazović | spheres and cells in computable metric spaces[END_REF][START_REF] Iljazovic | Compact manifolds with computable boundaries[END_REF][START_REF] Burnik | Computability of 1-manifolds[END_REF][START_REF] Iljazović | Semicomputable manifolds in computable topological spaces[END_REF][START_REF] Čičković | Chainable and circularly chainable semicomputable sets in computable topological spaces[END_REF][START_REF] Horvat | Computability of pseudo-cubes[END_REF][START_REF] Iljazović | Computability of graphs[END_REF][START_REF] Čelar | Computability of products of chainable continua[END_REF][START_REF] Čelar | Computability of glued manifolds[END_REF][START_REF] Djamel | Computability of finite simplicial complexes[END_REF] (see Fact 2.2.1) actually have strong computable type because the proofs hold relative to any oracle.

Note that the proof that the definition of computable type reduces to the Hilbert cube (Theorem 4.2.1) also extends to strong computable type, so defining strong computable type on computable metric spaces or computably Hausdorff spaces would yield equivalent notions.

We will see in Section 6.7.1 that for finite simplicial complexes, strong computable type is equivalent to computable type (Corollary 6.7.1). More generally, we expect that for natural spaces, the notion of strong computable type is actually no stronger than the notion of computable type.

Intuitively, a compact pair has strong computable type iff for every copy (Y, B) in the Hilbert cube, the set Y can be fully computed if we are only given the compact information about Y and B. We make it precise by seeing Y and B as points of the hyperspace of compact subsets of the Hilbert cube with suitable topologies.

The property of having strong computable type can then be rephrased as relative computability between elements of K(Q) with various topologies, using the notion of relative computability given by Definition 2.1.2. As relative computability is expressed in terms of enumeration reducibility, Definition 2.0.1 then provides equivalent formulations of this notion, which we will implicitly use in the rest of the thesis. Let us state such a formulation for clarity.

For a compact pair (X, A), the following statements are equivalent:

1. (X, A) has strong computable type, 2. For every copy (Y, B) of (X, A) in Q, the element Y of the space (K(Q), τ V ) is computable relative to the element (Y, B) of the product space (K 2 (Q), τ 2 upV ).

Proof. The second statement is about relative computability, which is defined as enumeration reducibility between neighborhood bases of Y and (Y, B). The definition of strong computable type is precisely formulation (3) of enumeration reducibility (Definition 2.0.1).

This formulation will imply useful consequences, because it enables one to use topology and descriptive set theory on K(Q) to analyze a computability-theoretic property.

A First Characterization

We give a first characterization of strong computable type. This result is related to the characterization obtained by Jeandel in [START_REF] Jeandel | Enumeration reducibility in closure spaces with applications to logic and algebra[END_REF] of the sets A ⊆ N that are total, i.e. whose complement N \ A is enumeration reducible to A.

We recall that (Y, B) is a proper subpair of (X, A), written (Y, B) (X, A), if Y X and B ⊆ A.

Definition 4.3.2. Let P be a property of compact pairs. A pair (X, A) ⊆ Q is P-minimal if (X, A) ∈ P and for every proper compact subpair (Y, B) (X, A), one has (Y, B) / ∈ P.

We can now state our first characterization of strong computable type.

Theorem 4.3.1. For a compact pair (X, A), the following statements are equivalent:

1. (X, A) has strong computable type, 2. For every copy (Y, B) of (X, A) in Q, there exists a property P of compact pairs which is Π 0 1 in the topology τ 2 upV and such that (Y, B) is P-minimal.

Proof. We fix an arbitrary copy of (X, A) in Q (that we still call (X, A)) and prove that the following equivalence holds for that copy: the set of basic τ V neighborhoods of X is enumeration reducible to the set of basic τ 2 upV neighborhoods of (X, A) iff there exists a Π 0 1 property P such that (X, A) is P-minimal.

2. ⇒ 1. Assume that P is Π 0 1 in τ 2 upV and that (X, A) is P-minimal. Given the τ 2 upV neighborhoods of (X, A), we need to enumerate the rational balls U intersecting X. Note that U intersects X iff X \ U is a proper subset of X; as (X, A) is P-minimal, it is equivalent to (X \ U, A \ U ) / ∈ P. Given (X, A) in the topology τ 2 upV , one can compute (X \ U, A \ U ) in the topology τ 2 upV (indeed, X \ U ⊆ V ⇐⇒ X ⊆ U ∪ V , and similarly for A \ U ) so one can semi-decide whether (X \ U, A \ U ) / ∈ P, i.e. whether U intersects X. 1. ⇒ 2. Consider a machine M which takes any enumeration of the basic τ 2 upV -neighborhoods of (X, A) and enumerates the open balls intersecting X. Let U be the set of compact pairs (X , A ) on which M fails in the following sense: after reading a finite sequence of τ 2 upV neighborhoods of the pair (X , A ), the machine enumerates an open ball U such that X ∩ U = ∅, where U is the corresponding closed ball. Let P be the complement of U. Note that (X, A) ∈ P because the machine does not fail on (X, A).

U (σ,U ) = {(X , A ) : ∀i ≤ k, (X , A ) ∈ σ i and X ⊆ Q \ U }.

It is an effective τ 2

upV -open set, and U is the union of all the U (σ,U ) such that M outputs U after reading σ, so it is a c.e. union. Therefore U is an effective τ 2 upV -open set and P is Π 0 1 in τ 2 upV . We now show that (X, A) is P-minimal. If (X , A ) is a proper compact subpair of (X, A), then there exists a ball U intersecting X such that U is disjoint from X . On an arbitrary enumeration of the τ 2 upV -neighborhoods of (X, A), the machine eventually outputs U after reading a finite sequence σ. As (X , A ) ⊆ (X, A), the τ 2 upV -neighborhoods of (X, A) are also τ 2 upV -neighborhoods of (X , A ) so the machine fails on (X , A ). Therefore, (X , A ) ∈ U, i.e. (X , A ) / ∈ P. We have shown that (X, A) is P-minimal.

In particular, this characterization immediately implies a simple sufficient condition for having strong computable type.

Theorem 4.3.2. Let P be a topological invariant which is Σ 0 2 in τ 2 upV . Every minimal element of P has strong computable type.

Proof. Let P = n P n \ Q n where each P n , Q n is Π 0 1 in τ 2 upV .
Assume that (X, A) is P-minimal and let (Y, B) be a copy of (X, A) in Q. This copy belongs to some

P n \ Q n . Let us show that (Y, B) must be P n -minimal. Let (Y , B ) be a proper compact subpair of (Y, B). The set Q n is Π 0 1 in τ 2 upV so it is an upper set. As (Y, B) / ∈ Q n , (Y , B ) / ∈ Q n as well. As (Y, B) is P-minimal, (Y , B ) / ∈ P, so (Y , B ) / ∈ P n .
Therefore, (Y, B) is P n -minimal. We have shown that each copy (Y, B) of (X, A) is P n -minimal for some n, so we can apply Theorem 4.3.1, implying that (X, A) has strong computable type.

We will see applications of this result in the sequel. Theorem 4.3.2 is only an implication and we will see later that the converse implication does not hold in general (see Section 5.4.4). However, in the next section we obtain a characterization of strong computable type using Σ 0 2 invariants and a weak form of minimality.

Discussion about Minimality

When applying Theorem 4.3.1 and Theorem 4.3.2 to prove that a pair has strong computable type, one needs to show that a pair (X, A) is P-minimal for some P. The definition of minimality involves all the proper compact subpairs (Y, B) (X, A). However, because P is Π 0 1 in τ 2 upV , it is not necessary to consider all of them.

The reason is that the lower and upper Vietoris topologies interact nicely with the inclusion ordering on compact sets:

If a set P ⊆ K(Q) is closed in τ upV or open in τ lowV , then it is an upper set: if K ⊆ K ⊆ Q and K ∈ P, then K ∈ P.
Dually, if P is open in the topology τ upV or closed in the topology τ lowV , then it is a lower set: if K ⊆ K ⊆ Q and K ∈ P, then K ∈ P.

It gives a shortcut to prove that a pair (X, A) is P-minimal, when P is an upper set, for instance when P is Π 0 1 or more generally closed in the topology τ 2 upV .

Definition 4.3.3. Let (X, A) be a compact pair and F a family of proper compact subpairs of (X, A). We say that F is cofinal if every proper compact subpair of (X, A) is contained in some pair in F.

The next result is elementary but very useful. Lemma 4.3.1

Let P be a property of compact pairs which is an upper set. Let (X, A) ∈ P and F be cofinal for (X, A). If no pair of F is in P then (X, A) is P-minimal.

Proof. If (Y, B) (X, A), then (Y, B) ⊆ (Z, C) (X, A) for some (Z, C) ∈ F, by cofinality. As (Z, C) / ∈ P by assumption and P is an upper set, (Y, B) / ∈ P.

We will implicitly apply this observation, notably with the following families:

The family

{(Y, Y ∩ A) : Y X} is cofinal, If B is

A Characterization using Invariants

In order to prove that a compact pair has strong computable type using Theorem 4.3.1, one needs to find a Π 0 1 property for each copy. In this section we improve the result by making the property less dependent on the copy. It is possible because there are countably many Π 0 1 properties, but uncountably many copies of a pair, so some property must work for "many" copies. The word "many" can be made precise using Baire category on the space of continuous functions of the Hilbert cube to itself.

Theorem 4.3.3. For a compact pair (X, A), the following statements are equivalent:

1. (X, A) has strong computable type, 2. There exists a Σ 0 2 invariant n∈N P n , where the sets P n are uniformly Π 0 1 in τ 2 upV , such that every copy of (X, A) in Q is P n -minimal for some n.

Note that the invariant is more than a Σ 0 2 property, because it is a union of closed sets rather than a union of differences of closed sets. The rest of the section is devoted to the proof of this result.

The idea of the proof is that one of the Π 0 1 properties from Theorem 4.3.1 works for "many" copies of (X, A), in the sense of Baire category. We then transform this Π 0 1 property into a Σ 0 2 invariant. This transformation is an instance of Vaught's transform [START_REF] Vaught | Invariant sets in topology and logic[END_REF]. Let us briefly explain this transform. There are several ways of defining an invariant out of a property P. For instance, the set of compact pairs having at least one copy in P is an invariant; the set of compact pairs whose copies are all in P is another invariant. However, these two invariants can have very high descriptive complexities (typically Σ 1 1 and Π 1 1 respectively). Vaught's transform is an intermediate way of converting a property into an invariant which almost preserves the descriptive complexity: the invariant is the set of compact pairs having "many" copies in P, where "many" is expressed using Baire category. We now give the details.

Note that we can make (C(Q), d) a computable metric space (see Definition 2.1.7).

Lemma 4.3.2

There exists a computable sequence of injective continuous functions

φ i : Q → Q that is dense in the metric d.
Proof. We first build a dense computable sequence of functions f i : Q → Q that are not necessarily injective. Say that an element x ∈ Q is dyadic of order n if it has the form x = (x 0 , . . . , x n-1 , 0, 0, 0, . . .) where each x i is a multiple of 2 -n . For each n ∈ N, the finite set of dyadic elements of order n forms a regular grid. One can then define a piecewise affine map by assigning a dyadic element to each dyadic element of order n and interpolating affinely in between.

All the possible such assignments provide a dense computable sequence of functions from Q to itself. Every continuous function f : Q → Q can be approximated by injective continuous functions g n : Q → Q defined as follows: let g n (x) be the sequence starting with the first n terms of f (x), appended with x. If f is computable then the functions g n are computable, uniformly. Therefore injective approximations of the functions f i give a computable dense sequence of injective functions.

Definition 4.3.4. Let > 0. An -function is a continuous function f : Q → Q sat- isfying d(f, id Q ) < . An -deformation of a pair (X, A) ⊆ Q is the image f (X, A) := (f (X), f (A)) of (X, A) under an -function f .
Note that if f is an -function and g is a δ-function, then f • g is an ( + δ)-function. Therefore, an -deformation of a δ-deformation of (X, A) is a ( + δ)-deformation of (X, A).

Remark 4.3.2

The notion of -function can be defined for partial functions f : X → Q, where X ⊆ Q, by requiring d X (f, id X ) < . The notion of -deformation of (X, A) does not change if we consider -functions define on Q or on X only, so we will freely use this flexibility in the sequel.

Indeed, if f : X → Q is an -function, then f has a continuous extension f : Q → Q which is an -function. The function f can be defined as follows: let f 0 : Q → Q be a continuous extension of f obtained by the Tietze extension theorem; let δ > 0 and f

(x) = (1 -λ x )f 0 (x) + λ x x, where λ x = min(1, d Q (x, X)/δ) (note that Q indeed allows convex combinations). f is a continuous extension of f and if δ is sufficiently small, then d( f, id Q ) < .
If a homeomorphism f : X → Y is an -function, then its inverse f -1 : Y → X is an -function as well.

Definition 4.3.5. Let P be a property of compact pairs. We define an invariant P * as follows: (X, A) ∈ P * iff there exists a copy (X , A ) of (X, A) and an > 0 such that everydeformation of (X , A ) is in P.

The key observation about P * is that its descriptive complexity is not too high, compared to the descriptive complexity of P:

if P is Π 0 1 , then P * is Σ 0 2 . Remark 4.3.3 When P is Π 0 1 (more generally closed) in τ 2
V , P * could be equivalently defined by considering injective -functions only, in which case -deformations of (X , A ) would all be copies of (X , A ). Indeed, the injective continuous functions are dense in C(Q), so every -deformation of (X , A ) is a limit, in the Vietoris topology, of -deformations by injective -functions; if the latter are all in P, then every -deformation is also in

P as P is closed in τ 2 V .
Proof of Theorem 4.3.3. For a rational > 0, we define

P = {(X, A) ⊆ Q : every -deformation of (X, A) is in P}.
One has P * = {(X, A) ⊆ Q : ∃ > 0, (X, A) has a copy in P }.

Claim 4.3.1 Let τ be either τ 2 V or τ 2 upV . If P is Π 0 1 in τ then P is Π 0 1 in τ . Proof. The function f ∈ C(Q) → f (X, A) ∈ (K 2 (Q), τ
) is continuous, P is closed in τ and the functions φ k are dense in C(Q). Therefore, the quantification over the -functions can be replaced by a quantification over the k's such that φ k is an -function: (X, A)

∈ P iff ∀k ∈ N, d(φ k , id Q ) < implies φ k (X, A) ∈ P.
It is a Π 0 1 predicate. Say that two pairs (X, A) and (Y, B) in Q are -homeomorphic if there exists a homeomorphism f : (X, A) → (Y, B) which is an -function (note that this relation is symmetric, because in this case f -1 is an -function as well). Claim 4.3.2

If (X 0 , A 0 ) ⊆ Q and (X 1 , A 1 ) ⊆ Q are homeomorphic, then for every > 0 there exists i such that φ i (X 0 , A 0 ) and (X 1 , A 1 ) are -homeomorphic.

Proof. Let f : (X 0 , A 0 ) → (X 1 , A 1 ) be a homeomorphism and let i be such that

d X 0 (φ i , f ) < . Let ψ = φ i • f -1 . One has φ i (X 0 , A 0 ) = ψ(X 1 , A 1 ) and d X 1 (ψ, id Q ) = d X 0 (φ i , f ) < . Claim 4.3.3 Let τ be either τ 2 V or τ 2 upV . If P is Π 0 1 in τ then P * is Σ 0 2 in τ .
Proof. Again, the idea is that we can replace the quantification over the copies, i.e. over the injective continuous functions, by a quantification over the φ i 's. Precisely, let

P i, = {(X, A) ⊆ Q : φ i (X, A) ∈ P }.
We show that P * = i, P i, , which is Σ 0 2 . If (X, A) has a copy (X , A ) in P , then there exists i such that φ i (X, A) ∈ P /2 . Indeed, let i be such that φ i (X, A) is /2-homeomorphic to (X , A ) by the previous claim. Every /2deformation of φ i (X, A) is an -deformation of (X , A ), so it belongs to P. Therefore φ i (X, A) ∈ P /2 , hence (X, A) ∈ P i, /2 . Claim 4.3.4

Assume that (X, A) is P-minimal and that (X, A) ∈ P . For every copy (X , A ) of (X, A), there exists i such that (X , A ) is P i, /2 -minimal.

Proof. Let (X , A ) be a copy of (X, A). Let i be such that φ i (X , A ) is /2-homeomorphic to (X, A). As in the previous claim, (X , A ) ∈ P i, /2 , and we show that (X , A ) is P i, /2 -minimal.

If (X , A ) is a proper compact subpair of (X , A ), then φ i (X , A ) is /2-homeomorphic to a proper compact subpair of (X, A). The latter is not in P as (X, A) was P-minimal, and is an /2-deformation of φ i (X , A ), so φ i (X , A ) / ∈ P /2 . In other words. (X , A ) / ∈ P i, /2 . As a result, (X , A ) is P i, /2 -minimal. If a compact pair (X, A) has strong computable type, then there exists a Π 0 1 property P, a copy (X 0 , A 0 ) ⊆ Q and an > 0 such that (X 0 , A 0 ) is P-minimal and (X 0 , A 0 ) ∈ P .

Proof. Assume that (X, A) ⊆ Q has strong computable type. Let I(Q) ⊆ C(Q) be the subspace of injective continuous functions f :

Q → Q. It is a G δ -set: I(Q) = n∈N I n (Q) where I n (Q) = {f ∈ C(Q) : ∀x, y ∈ Q, d(x, y) ≥ 2 -n =⇒ f (x) = f (y)}
is open by Proposition 2.1.1, because it is obtained as a universal quantification of an open predicate over the compact space Q × Q (see Lemma 1.3.10 in [START_REF] Van Mill | The Infinite-Dimensional Topology of Function Spaces[END_REF] for a more detailed proof). Being a G δsubset of the Polish space C(Q), the space I(Q) is therefore Polish (Theorem 3.11 in [START_REF] Kechris | Classical Descriptive Set Theory[END_REF]).

Theorem 4.3.1 implies that for each f ∈ I(Q) there exists a Π 0 1 property P (in τ 2 upV ) such that f (X, A) is P-minimal. There are countably many Π 0 1 properties, so by Baire category on I(Q) there exists a Π 0 1 -property P, an f 0 ∈ I(Q) and an > 0 such that for "almost every" f ∈ B(f 0 , ), f (X, A) is P-minimal. By "almost every", we mean for a dense set of functions f ∈ B(f 0 , ). It actually implies that for every continuous f ∈ B(f 0 , ) the pair f (X, A) is in P because P is closed and the injective functions are dense in the continuous ones by Lemma 4.3.2. The copy f 0 (X, A) belongs to P but may not be P-minimal. But, we can replace f 0 by any f 1 ∈ B(f 0 , /2) such that f 1 (X, A) is P-minimal, and by /2. We then let (X 0 , A 0 ) = f 1 (X, A).

Finally, we put everything together. Let (X 0 , A 0 ), P and be begin by Claim 4.3.5. The Σ 0 2 invariant we are looking for is P * = i, P i, . Claim 6.7.2 implies that every copy of (X, A) is P i, -minimal for some i, .

Strong Computable Type: Further Properties

In this section, we leverage the analysis conducted thus far to achieve a deeper comprehension of strong computable type from a structural perspective.

Product and Cone of Pairs

We give a relationship between pairs, their products and their cones, w.r.t. strong computable type.

The Product of Two Pairs Definition 4.4.1. The product of two pairs (X 1 , A 1 ) and (X 2 , A 2 ) is the pair

(X 1 , A 1 ) × (X 2 , A 2 ) = (X 1 × X 2 , (X 1 × A 2 ) ∪ (A 1 × X 2 )). Proposition 4.4.1 If (X 1 , A 1 ) × (X 2 , A 2 ) has (strong) computable type and (X 2 , A 2 ) has a semicomputable copy in Q, then (X 1 , A 1 ) has (strong) computable type. Proof. Assume that (X 1 , A 1 ) and (X 2 , A 2 ) are embedded in Q as semicomputable compact pairs, let us prove that X 1 is computable. The product (X, A) = (X 1 , A 1 ) × (X 2 , A 2 ) ⊂ Q × Q is semicom- putable because
each pair is. Therefore X is computable by assumption, so its first projection X 1 is computable. It shows that (X 1 , A 1 ) has computable type. The proof relativizes, so (X 1 , A 1 ) has strong computable type.

The Cone of a Pair

Proposition 4.4.2 Let (X, A) be a compact pair. If C(X, A) has (strong) computable type, then (X, A) has (strong) computable type.

Proof. We prove it for computable type and the proof relativizes. Suppose (Y, B) = C(X, A) has computable type and that (X 0 , A 0 ) is a semicomputable copy of (X, A) in Q. Let us prove that X 0 is computable.

Let

Y 0 = {(λ, λx) : λ ∈ [0, 1], x ∈ X 0 }, B 0 = {(λ, λx) : λ ∈ [0, 1], x ∈ A 0 } ∪ ({1} × X 0 ).
The pair (Y 0 , B 0 ) is a semicomputable realization of C(X 0 , A 0 ) in Q (because the pair (X 0 , A 0 ) is semicomputable and the function f

: x → {(λ, λx) : λ ∈ [0, 1]} and f (X 0 ) = Y 0 and f (A 0 ) = B 0 ) so Y 0 is computable by assumption. The intersection Z 0 of Y 0 with [1/2, 1]×Q is computable as well (an open set B intersects Z 0 iff the open set B ∩ ((1/2, 1] × Q) intersects Y 0 , which is semidecidable).
The image of Z 0 under the computable function (λ, y) → y/λ is X 0 , which is therefore computable. Here is another example. Let L be the so-called house with two rooms, or Bing's house (see Figure 4.2), and N = ∅. L is contractible and we proved in Proposition 6.7.2 that it has (strong) computable type. However, the results of Section 6.5.3 imply that as L is contractible, the pair (C(L), L) does not have strong computable type.

A Topological Necessary Condition

We establish a purely topological understanding of the concept of strong computable type through the identification of a necessary condition from a topological standpoint.

Corollary 4.4.1. (Necessary condition) If a compact pair (X, A) ⊆ Q has strong computable type, then there exists > 0 such that no sequence of -deformations of (X, A) converges to a proper compact subpair of (X, A) in the topology τ 2 upV .

Note that if A is empty one should not confuse between this necessary condition and the notion of strong approximation (see Definition 5.6.1).

Proof. From Theorem 4.3.1 and the proof of Theorem 4.3.3, (X, A) has strong computable type iff there exist a property of compact pairs P which is Π 0 1 in τ 2 upV , a copy (Y, B) of (X, A) in Q which is P-minimal, and an > 0 such that every -deformation of (Y, B) is in P.

As P is closed in τ 2

upV , every limit of -deformations of (Y, B) in that topology also belongs to P. As (Y, B) is P-minimal, no such limit is a proper compact subpair of (Y, B). The same result can be transferred to (X, A) by using a homeomorphism φ : (X, A) → (Y, B). Indeed, φ sends proper compact subpairs of (X, A) to proper compact subpairs of (Y, B) and if δ > 0 is sufficiently small, then φ sends δ-deformations of (X, A) to -deformations of (Y, B).

This necessary condition almost suffices and encapsulates the purely topological aspects of strong computable type. In fact, it is equivalent to the following relativization of strong computable type. Definition 4.4.2. A compact pair (X, A) has strong computable type relative to some oracle or relative strong computable type if there exists an oracle O such that for every copy (Y, B) of (X, A) in

Q one can compute Y in (K(Q), τ V ) from (Y, B) as an element of (K 2 (Q), τ 2
upV ) using O.

Corollary 4.4.2. For a compact pair (X, A), the following statements are equivalent:

1. (X, A) has strong computable type relative to some oracle, 2. There exists > 0 such that no sequence of -deformations of (X, A) converges to a proper compact subpair of (X, A) in the topology τ 2 upV .

Proof. The proof of Corollary 4.4.1 holds relative to any oracle, which shows (1) ⇒ (2). Now assume condition (2). Let P be the closure, in τ 2 upV , of the set of -deformations of (X, A). Let O be an oracle relative to which P is Π 0 1 in τ 2 upV . By definition, P contains all the -deformations of (X, A) and (X, A) is P-minimal by assumption. Therefore, we can apply the argument in the proof of Theorem 4.3.1, showing that (X, A) has strong computable type relative to O.

The contra-position of Corollary 4.4.1 gives a sufficient condition implying that a pair does not have strong computable type. In Section 4.5.1, we prove an effective version which even shows that the pair does not have computable type.

The -Surjection Property

We define three notions that characterize pairs. Namely, the generalized -surjection property, the -surjection property and the surjection property, we study them in details in Chapter 6.

Definition 4.4.3. Let > 0. A pair (X, A) ⊆ Q satisfies the generalized -surjection property if every continuous function of pairs f : (X, A) → (X, A) satisfying d X (f, id X ) < is surjective. The pair (X, A) ⊆ Q satisfies the -surjection property if every continuous function f : X → X satisfying d X (f, id X ) < and f | A = id A is surjective. X has the -surjection property if the pair (X, ∅) does. The pair has the surjection property if every continuous function f :

X → X such that f | A = id A is surjective.
The following special case of Corollary 4.4.1 has a particularly simple formulation.

Corollary 4.4.3. If (X, A) ⊆ Q has strong computable type, then it satisfies the generalized -surjection property for some > 0.

Proof. Take from Corollary 4.4.1. If f : (X, A) → (X, A) is an -function, then f (X, A) is andeformation of (X, A) which is a compact subpair of (X, A). By Corollary 4.4.1 it cannot be a proper compact subpair, which means that f must be surjective.

We do not know whether the converse of Corollary 4.4.3 holds, although we do not expect so. As we discuss now, it holds for particular pairs. Example 4.4.1

As already mentioned, we do not know whether the converse of Corollary 4.4.3 holds. However, there is a simple example showing that the -surjection property does not imply computable type in general. Consider the pair (X, A) in R 2 defined as follows: X = i∈N X i where X 0 = [0, 1]×{0} and for i ≥ 1,

X i = {2 -i } × [0, 2 -i ] and A = {(2 -i , 2 -i ) : i ≥ 1} ∪ {(0, 0)} (see Figure 4.3).
Every continuous function f : X → X such that f | A = id A must be surjective, so (X, A) satisfies the surjection property, hence the -surjection property for every . However it does not have computable type: if E ⊆ N is a non-computable c.e. set, then the pair (Y, B) defined by Y = X 0 ∪ i / ∈E X i and B = {(2 -i , 2 -i ) : i ≥ 1, i / ∈ E} ∪ {(0, 0)} is a semicomputable copy of (X, A) but Y is not computable.

Figure 4.3: A compact pair having the surjection property but not computable type

It is an open question whether there is a compact space with infinitely many connected components having strong computable type. The previous results give partial answers to this question. Essentially, such a space cannot have arbitrarily small connected components. Let X be a compact metric space.

If for every > 0 there exists a connected component of X of diameter < which is not a singleton, then X does not have strong computable type.

If X \ A contains infinitely many isolated points in (X, τ X ), then (X, A) does not have strong computable type.

Proof. We show that in both situations one can build for each > 0 a non-surjective continuousfunction, implying that X does not have strong computable type by Corollary 4.4.3.

Let A ⊆ X be a connected component of diameter < , which is not a singleton. A is the intersection of a decreasing sequence of clopen sets, so by compactness it is contained in a clopen set C of diameter < . The function f : X → X defined as the identity on X \ C and sending all C to some point x ∈ C is a continuous -function, which is not surjective because C is not a singleton.

If X \A contains infinitely many isolated points then by compactness, they have an accumulation point x ∈ X. Given > 0, let f : (X, A) → (X, A) send some isolated point y ∈ (X \ A) ∩ B(x, ) to x and be the identity on X \ {y}. f is a non-surjective continuous -function.

The first result in Proposition 4.4.3 may be generalized to pairs. However, the proof would not use the -surjection property because there exists pairs which have this property and do not have strong computable type.

For instance, take X to be a countable union of disjoint disks converging to set S disjoint from them and let A be the bounding spheres of the disks in addition to S. There exists a semicomputable copy of (X, A) relative to some oracle O which is not computable relative to it.

Indeed, enumerate each disk X i in X and let A i be its bounding sphere, take an oracle O which makes (X, A) semicomputable. If E ⊆ N is a non-computable c.e. set, then the pair (Y, B) defined by Y = S ∪ i / ∈E X i and B = S ∪ i / ∈E A i is a semicomputable copy of (X, A) relative to O but Y is not computable relative to it.

Uniformity

Given a compact pair (X, A) which has strong computable type, a natural question one may ask is whether it is possible to have a single effective procedure that takes any copy (Y, B) given in the topology τ 2 upV and computes Y in the topology τ V . It turns out that the answer is almost always negative: uniformity is only possible when X is a singleton (for a singleton X = {p} semicomputability is equivalent to computability because a ball B i contains p iff it intersects it, so there is a single effective procedure which will just output the semicomputability information since it equals the computability information), see Theorem 4.4.1. Therefore, the natural problem is to measure the degree of non-uniformity of this problem, using Weihrauch degrees. It was studied by Pauly [START_REF] Pauly | Weihrauch degrees of problems related to topological circles[END_REF] in the case of the circle embedded in R 2 . We give here more results about this question. Definition 4.4.4. For a compact pair (X, A), let SCT (X,A) be the function taking a copy (Y, B) of (X, A) in τ 2 upV and outputting Y in τ 2 V .

The first result holds for every compact pair (which may or may not have strong computable type). Definition 4.4.5. Closed choice over N is the problem C N of finding an element in a non-empty set A of natural numbers, given any enumeration of the complement of A.

Theorem 4.4.1. Let (X, A) be a compact pair such that X is not a singleton. One has C N ≤ t sW SCT (X,A) . If (X, A) has a semicomputable copy then C N ≤ sW SCT (X,A) .

Proof. Instead of C N , we use the strongly Weihrauch equivalent problem Max which sends a nonempty finite subset of N, represented by its characteristic function, to its maximal element (Theorem 7.13 in [START_REF] Brattka | Weihrauch complexity in computable analysis[END_REF]). We prove the result when (X, A) has a semicomputable copy, the general case is obtained by relativizing the argument to an oracle which semicomputes some copy.

Assume that (X, A) is already embedded as a semicomputable compact pair in Q. We can assume that 1/2 < diam(X) < 1, rescaling X if needed (and using the assumption that X is not a singleton).

Given a non-empty finite set E ⊆ N, let m = max E. We show how to produce a semicomputable copy (X E , A E ) of (X, A) such that 2 -m-1 < diam(X E ) < 2 -m . Given an access to X E in the topology τ V , one can compute its diameter so one can compute m, showing that Max is strongly Weihrauch reducible to SCT (X,A) .

We now define (X E , A E ). It is obtained from (X, A) by a translation and a scaling by a factor 2 -m . Given E, one can compute the sequence m i = max(E ∩ [0, i]) converging to m. The idea is that at stage i, we produce a copy of (X, A) that is translated and scaled by 2 -m i . If at stage i + 1, one has m i+1 = m i then we keep that copy. If m i+1 > m i , then we move to another copy. The new scaling factor is 2 -m i+1 and we choose the translation so that the new copy is contained in the current τ 2 upV neighborhood of the previous copy. It is possible because we can assume that at stage i, the τ 2 upV neighborhood contains a ball of diameter 2 -i , and 2 m i+1 = 2 -i-1 . In the other direction, we can prove that SCT (X,A) ≤ W C N holds when (X, A) has strong computable type in a particular way. A strong Weihrauch reduction is impossible, because C N has countably many possible outputs while SCT (X,A) has uncountably many. Theorem 4.4.2. Let P be a Σ 0 2 invariant in τ 2 upV . If a compact pair (X, A) is P-minimal then SCT (X,A) ≤ W C N .

Proof. As K(Q) is a metric space, P is of the form P = n P n where P n are uniformly Π 0 1 -sets in τ 2 upV . Given a copy (Y, B), let E = {n : (Y, B) ∈ P n }. From (Y, B) in τ 2 upV one can enumerate the complement of E, and given any n ∈ E, one can compute Y by using the fact that (Y, B) is P n -minimal as in the proof of Theorem 4.3.1.

We do not know whether SCT (X,A) is always Weihrauch reducible to C N when (X, A) has strong computable type. It is left as an open question (Question 6.8.2).

Discussion: Vietoris vs Upper Vietoris Topology

We now relate the descriptive complexity of properties of compact pairs in the Vietoris topology τ 2 V and upper Vietoris topology τ 2 upV . As τ 2 upV is weaker than τ 2 V , the descriptive complexity of a property in τ 2 upV is always an upper bound on its complexity in τ 2 V . Although the properties of a given complexity level are not the same in the topologies τ 2

V and τ 2 upV , we show that they actually induce the same class of minimal elements. Let P be a property of compact pairs. The set ↑P has the same minimal elements as P and

Notation. For a property of compact pairs

P let ↑P = {(X, A) ⊆ Q : ∃(X , A ) ⊆ (X, A), (X , A ) ∈ P}.
1. If P is Π 0 1 in τ 2 V , then ↑P is Π 0 1 in τ 2 upV , 2. If P is Σ 0 2 in τ 2 V , then ↑P is Σ 0 2 in τ 2 upV ; moreover, ↑P = n P n where P n are uniformly Π 0 1 in τ 2 upV .
Therefore, whether a compact pair (X, A) is minimal for some Π 0 1 or Σ 0 2 property does not depend on the topology (among τ 2

V and τ 2 upV ) in which the complexity is measured. In order to prove Proposition 4.4.4, we need the next result.

It is possible to prove more generally an effective contra-position of Corollary 4.4.1: if we assume that (X, A) ⊆ Q is semicomputable and that given > 0 one can compute δ > 0 such thatdeformations of (X, A) converge to a proper compact subpair (Y, B) (X, A) with d H (Y, X) > δ, then (X, A) does not have computable type. Note that neither the -deformations nor the limit Y are required to be computable. The computability assumption is only about the function sending to δ.

This result therefore identifies a general situation when computable type is equivalent to strong computable type. The statement given in Theorem 4.5.2 appearing in [6] is slightly stronger than the statement appearing in [START_REF] Djamel | Computability of finite simplicial complexes[END_REF]. Indeed, in [START_REF] Djamel | Computability of finite simplicial complexes[END_REF] the pair (X, A) is assumed to be computable. Moreover, the notion of witness defined here (and in [6]) is weaker than the one in [START_REF] Djamel | Computability of finite simplicial complexes[END_REF] (where f should be the identity on A). We have realized that this stronger result holds because the proof presented here is simpler and identifies more clearly the needed assumptions.

We now give a proof by applying Theorem 4.5.1. We assume that (X, A) is embedded as a semicomputable compact pair in Q which has computable witnesses. First, if X is not computable then (X, A) does not have computable type and the result is proved. Therefore, we can assume for the rest of the proof that X is computable (however, A may not be computable). Consider the space C(X, Q) of continuous functions from X to Q. It is endowed with a complete computable metric d(f, g) = max x∈Q d Q (f (x), g(x)), inducing a topology τ . The subspace I(X, Q) of injective continuous functions from X to Q is a dense Π 0 2 subset (complement of a Σ 0 2 set), in particular it contains a dense computable sequence. We consider two weaker topologies τ 1 and τ 2 on C(X, Q).

For each pair (U, V ) of finite unions of basic open subsets of Q, let

V U,V = {f ∈ C(X, Q) : f (X) ⊆ U, f (A) ⊆ V }
and let τ 1 be the topology generated by the sets V U,V as a subbasis.

For each basic open subset B of Q, let

U B = {f ∈ C(X, Q) : f (X) ∩ B = ∅}
and let τ 2 be the topology generated by the sets U B and V U,V as a subbasis. Our goal is to build an injective continuous function f ∈ C(X, Q) such that f (X) and f (A) are semicomputable but f (X) is not computable; in other words, we want f to be τ 1 -computable but not τ 2 -computable.

We will apply Theorem 4.5.1, so we need to show that τ 1 is τ -generially weaker than τ 2 .

Lemma 4.5.1

Let X ⊆ Q be a computable compact set and A be a semicomputable compact subset of X. If the pair (X, A) has computable witnesses, then the topology τ 1 is effectively generically τ -weaker than τ 2 .

Proof. We can assume that the centers of the basic metric balls in (C(X, Q), d) are injective functions. Given a metric ball B = B d (g 0 , ) in C(X, Q) (where g 0 and are computable and g 0 is injective), we need to compute B , B , U as in Definition 4.5.3. We are going to compute some suitable positive < and define:

B = B d (g 0 , ), B = {g ∈ C(X, Q) : d(g, g 0 ) < and d H (g(X), g 0 (X)) > }, U = {g ∈ C(X, Q) : d H (g(X), g 0 (X)) < }.
These sets are clearly effective open sets in the respective topologies. Note that B ⊆ B ∩ U and B ⊆ B \ U . We now explain how to choose so that B is contained in cl τ 1 (B ). Compute δ < /2 such that d Q (x, y) < δ implies d Q (g 0 (x), g 0 (y)) < /2. It implies that for all continuous functions g, h :

Q → Q, If d(h, g 0 ) < δ and d(g, id X ) < δ, then d(h • g, g 0 ) < . (4.1) Indeed, d(h • g, g 0 ) ≤ d(h • g, g 0 • g) + d(g 0 • g, g 0 ) < δ + /2 ≤ .
Compute β, a δ-witness for (X, A). Compute ≤ δ such that for all x, y ∈ X, d Q (g 0 (x), g 0 (y)) ≤ 2 implies d Q (x, y) ≤ β. It implies that for all non-empty compact sets Y, Z ⊆ X,

If d H (Y, Z) > β, then d H (g 0 (Y ), g 0 (Z)) > 2 . (4.2)
We now check that B is contained in cl τ 1 (B ). Let h ∈ B = B d (g 0 , ). As β is an δ-witness, there exists g : X → X such that g(A) ⊆ A, d(g, id X ) < δ and d H (X, g(X)) > β. We define g 1 = h•g and show that g 1 ∈ B . One has d(g 1 , g 0 ) < by (4.1), and

d H (g 1 (X), g 0 (X)) ≥ d H (g 0 (X), g 0 • g(X)) -d H (g 0 • g(X), h • g(X)) > 2 -d(g 0 , h) > by (4.2), so g 1 ∈ B . Moreover, g 1 (X) = h(g(X)) is contained in h(X) and g 1 (A) = h(g(A)) ⊆ h(A), so h belongs to cl τ 1 ({g 1 }) ⊆ cl τ 1 (B ). We have proved that B ⊆ cl τ 1 (B ).
Proof of Theorem 4.5.2.

The subset of injective continuous functions from X to Q is a τdense Π 0 2 (τ )-subset of C(X, Q). Therefore, applying Theorem 4.5.1, there exists an injective continuous function f : X → Q that is τ 1 -computable but not τ 2 -computable. In other words, the pair (f (X), f (A)) is semicomputable, but f (X) is not computable.

It may seem that using the effective Baire category theorem (see Theorem 2.1 in [6]) and Theorem 4.5.1 in [6] is a rather convoluted path to proving Theorem 4.5.2. A more direct proof is indeed possible (see [5]), but at the cost of readability, because there are many ingredients to take care of and to put together. These theorems isolate the appropriate concepts that make the construction possible, separating the specific properties of the application (Lemma 4.5.1) from the general construction the effective Baire category theorem (see Theorem 2.1 in [6]) and Theorem 4.5.1 in [6].

Simple copies

When a set or a pair lacks computable type, the task of constructing a semicomputable copy that is not computable can vary in terms of difficulty. For instance, it is straightforward to show that the line segment I does not have computable type (as a single set, i.e. without its boundary): if r > 0 is a non-computable right-c.e. real number, then [0, r] is a semicomputable copy of I which is not computable.

However, for other sets such as the dunce hat (see Definiton 6.7.1) or the set shown in Figure 4.4a below, there is no such obvious construction. We can formulate a precise statement expressing this idea, by using the results obtained so far, notably Theorem 4.3.1. Definition 4.5.5. Let X, Y be compact metric spaces and f : X → Y be continuous. A modulus of uniform continuity for f is a function µ : The set X shown in Figure 4.4a, consisting of a disk attached to a pinched torus, does not have computable type. It can be proved using the results in Section 6.7.1 and Section 4.5.1. Indeed, the neighborhood of the pinched point is the cone of a graph consisting of two circles attached by a line segment (see Proposition 6.7.1); the line segment is not part of a cycle in the graph, implying that the set does not have computable type. Another way to prove that it does not have computable type is to use the fact that for every > 0, there is a sequence of -deformations of X converging to a proper subset of X (see Figure 4.4b). Therefore, it does not have strong computable type by Corollary 4.4.1 (and in fact, it does not have computable type as it is a finite simplicial complex (Corollary 6.7.1).

N → N such that if d(x, x ) < 2 -µ(n) then d(f (x), f (x )) ≤ 2 -n .

An Example

As X does not contain a copy of itself, it has no simple copy which is semicomputable but not computable. It explains why the proof, given in [START_REF] Djamel | Computability of finite simplicial complexes[END_REF]6], that such sets do not have computable type is not straightforward.

A semicomputable copy of X which is not computable is illustrated in Figure 4.5. It could be obtained more directly by encoding the halting set: if (n i ) i∈N is a computable one-to-one enumeration of the halting set, then the holes appearing in the disk have sizes 2 -n 0 , 2 -n 1 , . . . etc.

Conclusion

I

n this chapter, we reduced the notion of computable type to the Hilbert cube and introduced the notion of strong computable type which is more flexible. We obtained characterizations of spaces which have strong computable type. In addition, we proved that being minimal satisfying some Σ 0 5. 6 The results stated in this chapter can be found in [START_REF] Djamel | Strong computable type. Computability[END_REF] and [START_REF] Djamel | Descriptive complexity of topological invariants[END_REF].

Introduction

W

hen analyzing the topology of a subset of the plane or a Euclidean space, several fundamental questions arise: How challenging is it to understand the topological properties of the set? Can we determine if it is homeomorphic to a specific space? How difficult is it to distinguish between two different spaces? More generally, what is the complexity involved in testing whether the set satisfies a particular topological invariant?

These questions can be approached in various ways, depending on the class of allowed spaces, the method of description, and the notion of complexity used to measure the difficulty of topological invariants.

In this chapter, our focus is on compact subsets of Euclidean spaces, and more generally compact subsets of the Hilbert cube [0, 1] N . Descriptions of such subsets essentially convey information about which voxels or rational balls intersect with them. However, these descriptions are limited in the sense that they do not provide structural or combinatorial details. For example, if the space is a topological graph, the description does not offer direct access to its combinatorial representation. As a result, no topological invariant can be determined in finite time. Therefore, the relevant measure of complexity is descriptive complexity, which quantifies the difficulty of expressing a property in terms of simpler properties. In its effective form, descriptive complexity measures the difficulty of expressing the property using a logical formula or a program.

The primary purpose of topological invariants is to distinguish between different topological spaces and demonstrate their non-homeomorphism. An invariant is considered interesting if it can effectively differentiate between spaces and contributes to a rich theoretical framework. From a logical or computational standpoint, an invariant is also valuable if it is straightforward to test.

In this chapter, our overarching goal is to understand which topological invariants can be tested or described using a limited level of complexity. We tackle the following specific problems: Characterize, to the extent possible, the topological invariants corresponding to a given level of complexity.

For a restricted class of spaces, particularly for pairs of spaces, determine the minimum level of complexity required to distinguish all the spaces in that class.

Given a specific space, ascertain the complexity involved in recognizing that space.

The study of descriptive complexity in topological invariants is particularly intriguing as it lies at the intersection of two branches of topology: point-set topology and algebraic topology. Descriptive complexity is part of point-set topology as it focuses on expressing sets of points in terms of open sets. Algebraic topology, on the other hand, offers topological invariants with low descriptive complexity.

While descriptive complexity of topological invariants has been explored in previous studies, our approach differs. Rather than determining the descriptive complexity of invariants provided by topology, we choose a complexity level and search for expressive invariants that possess that level of complexity. We investigate which spaces can be distinguished using invariants of that complexity level. Our particular emphasis is on invariants of low complexity, specifically Π 0 1 and Σ 0 2 invariants. We are motivated to study Σ 0 2 invariants because they play a crucial role in the study of the computability properties of topological spaces such as strong computable type (see Chapter 4).

In Section 5.2, we show how the classical notion of Absolute Neighborhood Retract (ANR) behaves well in terms of computability, and allows to define topological invariants of low descriptive complexity, expressing extensibility and null-homotopy of continuous functions to ANRs. In Section 5.3 we study these invariants in more details. We apply the theory in Section 5.4 to revisit many previous results about computable type. In Section 5.5, we study Π 0 1 invariants. In Section 5.6 we study Σ 0 2 invariants. In Section 5.7, we show how to separate finite topological graphs.

Computability-Theoretic Properties of ANRs

Theorem 4.3.2 gives a simple way to prove that a compact pair (X, A) has strong computable type by identifying a Σ 0 2 topological invariant for which (X, A) is minimal. In order to apply this result, we need to find suitable Σ 0 2 invariants. In this section, we develop results of independent interest that will be applied to define Σ 0 2 invariants, expressed in terms of extensions of continuous functions. At the heart of our framework lies the concept of an Absolute Neighborhood Retract (ANR), which holds significant importance in topology. Interestingly, ANRs possess valuable properties from a computability-theoretic perspective, and we will leverage these properties to define Σ 0 2 invariants. The general idea is that the space of functions from a space X to a space Y cannot be handled in a computable way if Y is arbitrary; it can if Y is an ANR.

We explore the computable aspects of ANRs. The results will be applied in the next section.

Absolute Neighborhood Retracts (ANRs)

We point out that computability-theoretic aspects of compact ANRs have been studied by Collins in [START_REF] Collins | Computability of homology for compact absolute neighbourhood retracts[END_REF], although we do not use these results.

The results we give here have important computability-theoretic consequences, because arbitrary functions can be replaced by computable functions that are close enough to the original ones so that they are homotopic.

Homotopy Classes

Let Y be a fixed compact ANR. Given a compact set X ⊆ Q in the upper Vietoris topology, we show that the set [X; Y ] can be computed in some way. Note that this question has been addressed, for instance in [START_REF] Čadek | Computing all maps into a sphere[END_REF], when X and Y are simplicial complexes presented as combinatorial objects. In our case, the information available about X and Y is much more elusive, therefore one might expect that very little can be computed. Note that computable aspects of ANRs have been investigated by Collins [START_REF] Collins | Computability of homology for compact absolute neighbourhood retracts[END_REF], where a compact ANR Y is presented together with a neighrborhood and a computable retraction. In our case, we do not have access to a retraction.

In order to state the results, let us discuss the consequences of Borsuk's homotopy extension theorem (Theorem 3.3.1). Let (X, A) be a compact pair. Whether a continuous function f : A → Y has a continuous extension F : X → Y depends on its equivalence class in [A; Y ]. We denote this class by [f ] A to make it clear that f is defined on A and that we consider homotopies defined on A (rather than X). If f : A → Y and F : X → Y are continuous functions, then we say that [F ] X extends [f ] A if F is homotopic to an extension of f . This relation is well-defined, i.e. does not depend on the representatives of the equivalence classes, because if

F extends f , [g] A = [f ] A

In particular, [F ] X extends [f ] A if and only if [F |

A ] A = [f ] A .
A numbering of a set S is a surjective partial function ν S :⊆ N → S.

Theorem 5.2.1. Let Y be a fixed compact ANR. To any compact set X ⊆ Q we can associate a numbering ν [X;Y ] of [X; Y ] such that:

1. dom(ν [X;Y ] ) is c.e. relative to X, 2. Equality {(i, j) ∈ dom(ν [X;Y ] ) 2 : ν [X;Y ] (i) = ν [X;Y ] (j)} is c.e. relative to X,
3. For any compact pair (X, A), the extension relation where X and A are given as elements of (K(Q), τ upV ).

{(i, j) ∈ dom(ν [X;Y ] ) × dom(ν [A;Y ] ) : ν [X;Y ] (i) extends ν [A;Y ] (j)} is c.e.
Note that the result holds with no computability assumption about Y . We now proceed with the proof of this result.

Lemma 5.2.1

If Y ⊆ Q is a compact ANR, then there exist an effective open set U containing Y , a retraction r : U → Y and µ, α > 0 such that:

1. Functions to Y that are α-close are homotopic,

2. For x ∈ U , d Q (r(x), x) < µ, 3. For x, y ∈ U , d Q (x, y) < µ implies d Q (r(x), r(y)) < α.
Proof. The number α exists by Lemma 3.3.2. As Y is an ANR, there exists an open set W containing Y and a retraction r :

W → Y . Let V be an open set such that Y ⊆ V ⊆ V ⊆ W . As V is compact, r is uniformly continuous on V so there exists µ > 0 satisfying (iii) for x, y ∈ V . As r| Y = id Y and Y is compact, if x is sufficiently close to Y , d Q (r(x), x) < µ.
Therefore we can choose U ⊆ V satisfying 2. and by compactness of Y , U can be replaced by a finite union of rational balls.

Proof of Theorem 5.2.1. Let (f i ) i∈N be a dense computable sequence of functions f i : Q → Q. Let Y ⊆ Q be a compact ANR and let U, r, α, µ be provided by Lemma 5.2.1. For X ⊆ Q, we define a numbering ν [X;Y ] of [X; Y ] and prove (1), ( 4), ( 2) and (3) in this order. Its domain is

dom(ν [X;Y ] ) = {i ∈ N : f i (X) ⊆ U },
which is clearly c.e. relative to X, showing (1). For i ∈ dom(ν [X;Y ] ), we then define ν [X;Y ] (i) as the equivalence class of r

• f i | X : X → Y . We first show that ν [X;Y ] is surjective. For a continuous function f : X → Y , let f : Q → Q be a continuous extension of f . If f i is sufficiently close to f , then f i (X) ⊆ U (hence i ∈ dom(ν [X;Y ] )) and r • f i | X is α-close to f . Therefore, r • f i | X is homotopic to f and ν [X;Y ] (i) = [f ].
Let now (X, A) be a compact pair in Q and let ν [X;Y ] and ν [A;Y ] be the corresponding numberings. We show condition (4). Proof of the claim. We show that r

• f i | A : A → Y has a continuous extension to X iff there exists j ∈ dom(ν [X;Y ] ) such that d A (f j , f i ) < µ.
It will imply the result as this condition is c.e. relative to X and A.

If there exists such a j,

then d A (r • f j , r • f i ) < α by Lemma 5.2.1 (iii) so r • f j | A and r • f i | A are homotopic by Lemma 5.2.1 (i). As r • f j | A obviously has an extension r • f j | X , so does r • f i | A by Borsuk's homotopy extension theorem (see Theorem 3.3.1).
Conversely, assume that r

•f i | A has an extension f : X → Y . One has d A (f, f i ) = d A (r •f i , f i ) < µ by Lemma 5.2.1 (ii) so if f j is sufficiently close to f on X, then f j (X) ⊆ U and d A (f j , f i ) < µ.
We now show (2), i.e. that equality is c.e. Let Cyl(X) = [0, 1] × X ⊆ Q be the cylinder of X and ∂Cyl(X) = {0, 1} × X its boundary. These sets are semicomputable relative to X. A pair (f, g) of functions from X to Y is nothing else than a function h : ∂Cyl(X) → Y , defined as h(x, 0) = f (x) and h(x, 1) = g(x). This correspondence is moreover effective, in the sense that there exists a computable function ϕ : dom(ν [X;Y ] ) → dom(ν [∂Cyl(X);Y ] ) such that if i, j are indices of the homotopy classes of f and g respectively, then ϕ(i, j) is an index of the homotopy class of h. Note that and f, g are homotopic iff h has a continuous extension to Cyl(X). Therefore, for i, j ∈ dom(ν [X;Y ] ), one has ν [X;Y ] (i) = ν [X;Y ] (j) iff ν [∂Cyl(X);Y ] (ϕ(i, j)) has an extension to Cyl(X), which is c.e. relative to X by Claim 5.2.1 applied to the pair (Cyl(X), ∂Cyl(X)).

We finally show (3), i.e. the extension relation is c.e. Note that dom(ν

[X;Y ] ) ⊆ dom(ν [A;Y ] ) and for i ∈ dom(ν [X;Y ] ), ν [A;Y ] (i) is the equivalence class of the restrictions to A of functions in ν [X;Y ] (i). For i ∈ dom(ν [X;Y ] ) and j ∈ dom(ν [A;Y ] ), ν [X;Y ] (i) extends ν [A;Y ] (j) iff ν [A;Y ] (i) = ν [A;Y ] (j), which is c.e.

Families of Σ 0 2 Invariants

The previous results enable us to define families of Σ 0 2 invariants. We will study their properties and use particular instances in the next sections.

1) Extension of Functions

To each compact ANR Y we associate a topological invariant E(Y ). Ultimately, we will focus on Y = S n , the n-dimensional sphere.

Definition 5.2.1. (The invariant E(Y ))

Let Y be a topological space. A compact pair (X, A) belongs to E(Y ) if there exists a continuous function f : A → Y that cannot be extended to a continuous function g : X → Y .

Note that this definition is only interesting when A is non-empty, otherwise it is never satisfied.

From this definition, the obvious upper bound on the descriptive complexity of E(Y ) is Σ 1 2 . However when Y is a compact ANR, we show that the complexity drops down to Σ 0 2 , which is much lower and is an effective class, even if there is no computability assumption on Y .

Corollary 5.2.1. If Y is a compact ANR, then E(Y ) is a Σ 0 2 invariant in τ 2 upV .
Proof. It is a simple application of Theorem 5.2.1. Indeed,

(X, A) ∈ E(Y ) ⇐⇒ ∃f : A → Y having no extension to X, ⇐⇒ ∃i ∈ dom(ν [A;Y ] ), ∀j ∈ dom(ν [X;Y ] ), ν [X;Y ] (j) does not extend ν [A;Y ] (i)
which is a Σ 0 2 condition. Note that although E(Y ) is a countable union of differences of closed sets, it is never a countable union of closed sets in τ 2 upV unless it is empty. Indeed, if E(Y ) is non-empty then it is not an upper set: take (X, A) ∈ E(Y ), and observe that (X, X) / ∈ E(Y ).

2) Null-Homotopy of Functions

The previous results provide another family of Σ 0 2 invariants.

Definition 5.2.2. (The invariant H(Y ))

Let Y be a topological space. A space X belongs to H(Y ) if there exists a continuous function f : X → Y which is not null-homotopic, i.e. not homotopic to a constant function.

Corollary 5.2.2. If Y is a compact ANR, then H(Y ) is a Σ 0 2 invariant in τ 2 upV . Proof. One has X ∈ H(Y ) iff [X; Y ] is non-trivial iff ∃i, j ∈ dom(ν [X;Y ] ), ν [X;Y ] (i) = ν [X;Y ] (j),
which is a Σ 0 2 condition. Slightly differently, we fix some i 0 such that

f i 0 : Q → Q is constant with value in U (implying i 0 ∈ dom(ν [X;Y ] ) for any X), so that [X; Y ] is non-trivial iff ∃i ∈ dom(ν [X;Y ] ), ν [X;Y ] (i) = ν [X;Y ] (i 0 ).
Again, H(Y ) is not a countable union of closed sets in τ upV unless it is empty, because it is not an upper set in that case: the maximal set Q is not in H(Y ) because it is contractible.

The invariant H(Y ) only applies to single sets. We will define a version for compact pairs, with Y = S n .

Detailed study of some Σ 0 2 Invariants

We now study in details the Σ 0 2 invariants defined in the previous section.

Properties of E(Y )

We study the invariant E(Y ) from Definition 5.2.1, and develop a few techniques to establish whether a compact pair satisfies this invariant. First, if A is a retract of X, then (X, A) / ∈ E(Y ). Indeed, every f : A → Y has an extension f •r : X → Y , where r : X → A is a retraction.

The next result identifies a relation between compact pairs that preserves the invariant E(Y ).

Proposition 5.3.3 Whether f i : X i → Y are asymptotically homotopic to f : X → Y does not depend on the choice of a neighborhood U of X and an extension f : U → Y .

Proof. Assume that for some U and f : U → Y and sufficiently large i, f i is homotopic to f | X i . Let V be a neighborhood of X and g : V → Y a continuous extension of f . As f and g coincide on X, there exists > 0 such that their restrictions f | N (X) and g| N (X) are as close as needed so that they are homotopic. For sufficiently large i, f i is homotopic to f | X i by assumption and

X i is contained in N (X), so f | X i is homotopic to g| X i . Therefore, f i is homotopic to g| X i .
Now, we prove that asymptotic homotopy preserves continuous extensibility.

Proposition 5.3.4

Let Y be an ANR, let (X i , A i ) i∈N and (X, A) be compact pairs in Q such that (X i , A i ) converge to (X, A) in the upper Vietoris topology. Assume that f i : A i → Y are asymptotically homotopic to f : A → Y . If f has a continuous extension on X, then for sufficiently large i, f i has a continuous extension on X i .

Proof. Let F : X → Y be a continuous extension of f . It has a continuous extension F : U → Y for some neighborhood U of X. Note that U is a neighborhood of A and F is an extension of f . As f i is asymptotically homotopic to f , f i is homotopic to F | A i for large i. The function F | A i has a continuous extension to X i , namely F | X i , so f i also has a continuous extension as well.

In particular, when all the involved pairs have the same second component, we obtain the following simple result.

Corollary 5.3.1. Let Y be an ANR. Let (X i , A i ) i∈N and (X, A) be compact pairs in Q such that A i = A for all i and X i converge to X in the upper Vietoris topology. If f : A → Y has no continuous extension to X i for any i ∈ N, then f has no continuous extension to X.

The Invariant E n

For n ∈ N, the n-dimensional sphere S n is an ANR. We define E n = E(S n ). This invariant will be used to cover several examples. The simplest example of a compact pair which is E n -minimal is given by the n + 1-ball and its bounding n-sphere. We first explain why it is in E n . More generally, whether a pair (B m+1 , S m ) satisfies E n depends on the mth homotopy group of the n-sphere. Proposition 5.3.5

The pair (B m+1 , S m ) is in E n if and only if the homotopy group π m (S n ) is non-trivial.

In particular, (B n+1 , S n ) is in E n and (B m+1 , S m ) is not in E n for m < n.

Proof. As B m+1 is the cone of S m , a continuous extension of f : S m → S n to B m+1 is nothing else than a null-homotopy of f .

Relationship with Dimension

The dimension of a space has a strong impact on the possibility of extending functions to a sphere. First, if X has dimension at most n, then any continuous function from a closed subset to S n extends continuously to X (see Theorem 3.6.3 in [60].

Theorem 5.3.1. Let n ∈ N. If (X, A) is a compact pair with dim(X \ A) ≤ n, then (X, A) / ∈ E n . In particular, if dim(X) ≤ n then (X, A) / ∈ E n .
In some cases, if a compact pair is not in E n then no compact subpair is in E n .

Lemma 5.3.1

Let (X, A) be a compact pair with

dim(A) ≤ n. If (X, A) / ∈ E n , then for every (Y, B) ⊆ (X, A) one has (Y, B) / ∈ E n .
Proof. As dim(A) ≤ n, one has (A, B) / ∈ E n . Therefore, (X, A) / ∈ E n implies (Y, B) / ∈ E n : every function f on B has an extension to A and then to X, whose restriction to Y is an extension of f . This result gives a simple way to prove E n -minimality in certain circumstances.

Corollary 5.3.2. Let (X, A) be a pair in E n , where A has empty interior and dim(A) ≤ n.

If for every A ⊆ Y X one has (Y, A) / ∈ E n , then (X, A) is E n -minimal.
Proof. Let (Y, B) be a proper compact subpair of (X, A). As A has empty interior, For every n ∈ N, the pair

Y ∪ A is a proper subset of X, so (Y ∪ A, A) / ∈ E
(B n+1 , S n ) is E n -minimal. Proof. By Proposition 5.3.5, (B n+1 , S n ) is in E n . If S n ⊆ Y B n+1 , then Y retracts to S n so (Y, S n ) / ∈ E n . Therefore by Corollary 5.3.2, no proper compact subpair (Y, B) of (B n+1 , S n ) is in E n . As a result, (B n+1 , S n ) is E n -minimal.
Therefore, our results give an alternative proof that the pair (B n+1 , S n ) has (strong) computable type, which was shown by Miller [START_REF] Miller | Effectiveness for embedded spheres and balls[END_REF]. We will give more examples of E n -minimal pairs in the next sections.

The Invariant H n

It was shown in [START_REF] Iljazovic | Compact manifolds with computable boundaries[END_REF][START_REF] Iljazović | Semicomputable manifolds in computable topological spaces[END_REF] that closed manifolds and compact manifolds with boundary have computable type. In this section, we identify a Σ 0 2 invariant for which manifolds are minimal. Let p ∈ S n be an arbitrary point in the n-sphere. We recall that two continuous functions of pairs f, g : (X, A) → (S n , p) are homotopic relative to A if there exists a homotopy between f and g that is constant on A, i.e. a continuous function

H : [0, 1] × X → S n such that H(0, x) = f (x), H(1, x) = g(x)
for all x ∈ X, and H(t, a) = p for all t ∈ [0, 1] and a ∈ A. A continuous function f : (X, A) → (S n , p) is null-homotopic relative to A if f is homotopic, relative to A, to the constant function p. Definition 5.3.3. (The invariant H n ) A compact pair (X, A) is in H n if there exists a continuous function of pairs f : (X, A) → (S n , p) which is not null-homotopic relative to A. A space X is in H n if the pair (X, ∅) is in H n , i.e. if there exists a continuous function f : X → S n which is not null-homotopic. Now that for single spaces, H n is precisely H(S n ) from Definition 5.2.2. We first prove that H n is a Σ 0 2 invariant, by reducing it to E n . Proposition 5.3.7

For a compact pair (X, A), one has

(X, A) ∈ H n ⇐⇒ C(X, A) ∈ E n . Moreover, if C(X, A) is E n -minimal, then (X, A) is H n -minimal. Proof. Let (C, D) = C(X, A) = (C(X), X ∪ C(A)).
A continuous function f : (X, A) → (S n , p) has a canonical extension F : D → S n with constant value p on C(A). Moreover, a null-homotopy of f relative to A is nothing else than a continuous extension of F to C.

Assume that (X, A) ∈ H n and let f : (X, A) → (S n , p) be a continuous function which is not nullhomotopic relative to A. By the previous observation, the function

F : D → S n has no continuous extension to C, therefore C(X, A) ∈ E n .
Conversely, assume that C(X, A) ∈ E n and let G : D → S n have no continuous extension to C. We cannot directly apply the preliminary observation to G because it is not necessarily constant on C(A). However, G is homotopic to a function F : D → S n which is constant on C(A). Indeed, C(A) is contractible so the restriction of G to C(A) is null-homotopic. As S n is an ANR, the homotopy on C(A) can be extended to a homotopy between G : D → S n and a function F : D → S n which is constant on C(A) (Theorem 3.3.1). Let p be the constant value. As G has no extension to C, neither does F . We can now apply the preliminary observation: the restriction f : X → S n of F to X is not null-homotopic relative to A, therefore (X, A) ∈ H n .

Assume that C(X, A) is E n -minimal. If (X , A ) is a proper compact subpair of (X, A), then the pair C(X , A ) is a proper compact subpair of C(X, A) so it does not satisfy E n , therefore (X , A ) / ∈ H n .

We will see below (Proposition 5.3.9) that under additional assumptions, the implication about minimality can be turned into an equivalence.

This characterization allows one to derive results about H n from results about E n . We start with the descriptive complexity of H n .

Corollary 5.3.3. H n is a Σ 0 2 invariant in τ 2 upV .
Proof. Given a compact pair (X, A) ⊆ Q, a copy (C, D) of C(X, A) can be defined as follows:

C = {(t, tx) : t ∈ [0, 1], x ∈ X}, D = {(1, x) : x ∈ X} ∪ {(t, tx) : t ∈ [0, 1], x ∈ A}.
The map φ sending (X, A)

∈ (K 2 (Q), τ 2 upV ) to (C, D) ∈ (K 2 (Q), τ 2
upV ) is easily computable, and H n is the preimage of E n by φ by Proposition 5.3.7. As E n is Σ 0 2 , so is H n .

As H n can be expressed using E n , the relations between E n and the dimension of the space can be similarly formulated for H n . Theorem 5.3.1 and Lemma 5.3.1 have the following consequences.

Proposition 5.3.8

Let (X, A) be a compact pair with dim(X) ≤ n -1. One has (X, A) / ∈ H n .

Proof. One has dim(C(X)) ≤ n so C(X, A) / ∈ E n by Theorem 5.3.1, therefore (X, A) / ∈ H n .

Lemma 5.3.2

Let (X, A) be a compact pair satisfying dim(X) ≤ n and dim(A

) ≤ n -1. If (X, A) / ∈ H n , then for every compact pair (Y, B) ⊆ (X, A) one has (Y, B) / ∈ H n .
Proof. We apply Lemma 5.3.1 to the pairs (C, D) = C(X, A) and (E,

F ) = C(Y, B). Note that D = X ∪ C(A) so dim(D) ≤ n. Therefore, if (C, D) / ∈ E n , then (E, F ) / ∈ E n .
Under the same assumptions, we can improve Proposition 5.3.7 to obtain an equivalence.

Proposition 5.3.9

Let (X, A) be a compact pair where A has empty interior, dim(X) ≤ n and dim(A

) ≤ n -1. One has (X, A) is H n -minimal ⇐⇒ C(X, A) is E n -minimal.
Before proving this result, we need to reformulate Borsuk's homotopy extension theorem (Theorem 3.3.1). The cylinder of a space X is Cyl(X) = X × [0, 1]. A homotopy H : [0, 1] × X → Y is nothing else than a function from Cyl(X) to Y , and a null-homotopy is a function from C(X) to Y . Therefore, Borsuk's homotopy extension theorem can be reformulated as follows. Proof. The first statement is Borsuk's homotopy extension theorem. The second one is a particular case for functions that are constant on X. More precisely, C(A) is the quotient of X ∪ Cyl(A) by identifying all the points of X. A function g : C(A) → Y induces a function f : X ∪ Cyl(A) → Y which is constant on X, by composing g with the quotient map. f has a continuous extension F : Cyl(X) → Y , which is constant on X, and therefore induces a continuous function G : C(X) → Y extending g.

Proof of Proposition 5.3.9. Assume that (X, A) is H n -minimal. We need to show that no proper compact subpair (Y, B) of C(X, A) is in E n . Using Corollary 5.3.2, it is sufficient to show the result when B = X ∪ C(A). Let then Y be a proper subset of C(X) containing X ∪ C(A). There exists a non-empty open set U ⊆ X \ A and an interval J = (a, b) ⊆ [0, 1] such that U × J is disjoint from Y (we think of U × J as embedded in C(X) in the obvious way). Let f : X ∪ C(A) → S n . We show how to extend f to C(X) \ (J × U ), giving an extension of f to Y by restriction. Figure 5.1 may help visualize the argument.

As (X, A) is H n -minimal by assumption, (X \ U, A) / ∈ H n so C(X \ U, A) / ∈ E n . Therefore, the restriction of f to (X \ U ) ∪ C(A) has a continuous extension to C(X \ U ). So f has a continuous extension g :

X ∪ C(X \ U ) → S n .
Applying Lemma 5.3.3, we can extend g to the quotient of X × [0, a] which is homeomorphic to C(X) and to Cyl(X) := X × [b, 1]. So we obtain a continuous extension h of f to the union of C(X \ U ) with the quotient of X For every pair (X, A), (X, A) ∈ H n ⇐⇒ X/A ∈ H n .

× ([0, a] ∪ [b, 1]). That union is precisely C(X) \ (J × U ). X A (a) f is defined on X ∪ C(A) X A U X ∪ Cyl(X \ U ) { Cone(X \ U ) { (b) g is defined on X ∪ C(X \ U ) X A U 0 1 a b Cyl(X) { Cone(X) { (c) h is defined on C(X) \ (U × J)
Proof. Let p A ∈ X/A be the equivalence class of A. First observe that (X, A) can equivalently be replaced by (X/A, p A ), i.e. (X, A) ∈ H n ⇐⇒ (X/A, p A ) ∈ H n . Indeed, there is a one-to-one correspondence between continuous functions f : (X, A) → (S n , p) and continuous functions g : (X/A, p A ) → (S n , p), and a one-to-one correspondence between null-homotopies of f relative to A and null-homotopies of g relative to p A . Now, for a pair (Y, q), where q is a point of Y , we show that a function g : (Y, q) → (S n , p) is null-homotopic if and only if it is null-homotopic relative to q. Assume that g is null-homotopic, i.e. homotopic to a constant function. We can assume w.l.o.g. that the constant function is p, because all constant functions are homotopic (indeed, the sphere is path-connected). Let H t be a homotopy from H 0 = g to H 1 = p. Our goal is to define another such homotopy H which is constantly p on q.

We give a different argument for n = 1 and n ≥ 2.

Let n = 1. There exists a continuous function φ : S 1 × S 1 → S 1 such that φ(x, p) = x and φ(x, x) = p for all x ∈ S 1 . Indeed, identifying S 1 with R/Z and p with the equivalence class of 0, let φ(x, y) = x -y mod 1. We can then define H t (y) = φ(H t (y), H t (q)), it is easy to check that it satisfies the required assumptions.

Let now n ≥ 2 (we do not know whether a similar map φ : S n × S n → S n can be defined but we provide another argument, which cannot be applied to n = 1). We want to define a continuous function

H : [0, 1] × Y → S n extending a certain function h : ([0, 1] × {q}) ∪ ({0, 1} × Y ) → S n .
Let h be the restriction of H to that set. We show that h is homotopic to h. As h has a continuous extension, it will imply that h has a continuous extension, because S n is an ANR.

On [0, 1] × {q}, h and h define two loops on S n (h defines the constant loop). As S n is simplyconnected (indeed, n ≥ 2), these two loops are homotopic relative to {0, 1} × {q}. As h and h coincide on {0, 1} × Y , they are homotopic.

We conclude this section with two simple examples of compact pairs and sets that are H n -minimal. Proposition 5.3. 11The pair (B n , S n-1 ) and the space S n are H n -minimal.

Proof. It is a direct application of Proposition 5.3.7: one has C(B n , S n-1 ) = C(S n , ∅) = (B n+1 , S n ) which is E n -minimal by Proposition 5.3.6.

We will see in the Section 5.4.2 that, more generally, compact n-dimensional manifolds with or without boundary are all H n -minimal.

Connections with Homology and Cohomology

Concluding this section, we present a partial characterization of E n and H n in terms of homology and cohomology groups for specific spaces. While our aim is not to provide a comprehensive exposition, we want to emphasize the close connections with algebraic topology that can be utilized in the study of strong computable type, particularly in the context of manifolds.

1) Partial Characterization of E n

For compact spaces of dimension at most n + 1, E n can be characterized in terms of Čech homology and cohomology (Corollaries V III.2 and V III.3 (p. 149) in [START_REF] Hurewicz | Dimension Theory (PMS-4)[END_REF]). For a pair (X, A), let i : A → X be the inclusion map. Like any continuous map, it induces natural homomorphisms on the homology and cohomology groups, written i * : Ȟn (A; G) → Ȟn (X; G) and i * : Ȟn (X; G) → Ȟn (A; G). It uses the abelian groups (Z, +) and (R/Z, +). Theorem 5.3.2. Let (X, A) be a compact pair with dim(X) ≤ n + 1. One has

(X, A) ∈ E n ⇐⇒ i * : Ȟn (A; R/Z) → Ȟn (X; R/Z) is not injective ⇐⇒ i * : Ȟn (X; Z) → Ȟn (A; Z) is not surjective.
The equivalence does not hold in general. For instance, Hopf's fibration is a continuous function f : S 3 → S 2 which is not null-homotopic, hence does not extend to B 4 , implying that (B 4 , S 3 ) ∈ E 2 . However, the second homology and cohomology groups of both B 4 and S 3 are trivial, so the only homomorphism between them is bijective.

2) Partial Characterization of H n

The invariant H n also admits a characterization in terms of Čech homology and cohomology, for compact spaces of dimension at most n (Corollary 4 (p. 150) in [START_REF] Hurewicz | Dimension Theory (PMS-4)[END_REF]).

Theorem 5.3.3. Let X be a compact space with dim(X) ≤ n. One has

X ∈ H n ⇐⇒ Ȟn (X; R/Z) = 0 ⇐⇒ Ȟn (X; Z) = 0.
Again, the equivalence does not hold in general. Hopf's fibration implies that

S 3 ∈ H 2 , but Ȟ2 (S 3 ; R/Z) ∼ = Ȟ2 (S 3 ; Z) ∼ = 0.
Lupini, Melnikov and Nies [START_REF] Lupini | Computable topological abelian groups[END_REF], and Melnikov and Downey [START_REF] Downey | Effectively compact spaces[END_REF] with a different proof, proved that if X ⊆ Q is given in the Vietoris topology τ V , then its Čech cohomology groups Ȟn (X; Z) can be computably presented (relative to X and uniformly): one can enumerate a list of generators and the words, i.e. finite combinations of generators and their inverses, that are equal to the 0 element of the group. Their result implies that the non-triviality of a group Ȟn (X; Z) is a Σ 0 2 invariant in the topology τ V : the group is non-trivial iff there exists a word which does not equal 0, which is a Σ 0 2 formula. It seems that their proof even shows that it is a Σ 0 2 invariant in the topology τ upV .

However Proposition 4.4.4 implies that being Σ 0 2 in τ V is enough for the purpose of proving strong computable type. More precisely, Corollary 5.3.4. Let X be a compact space. If X is minimal such that Ȟn (X; Z) 0, then X has strong computable type.

Proof. The invariant

I n = {X ∈ K(Q) : Ȟn (X; Z) 0} is Σ 0
2 in the topology τ V by [START_REF] Lupini | Computable topological abelian groups[END_REF] and [START_REF] Downey | Effectively compact spaces[END_REF]. Therefore, ↑I n is Σ 0 2 in the topology τ upV and has the same minimal elements as I n (Proposition 4.4.4), so we can apply Theorem 4.3.2.

We also mention a stronger connection between maps to the sphere and cohomology. Theorem 5.3.3 is actually a particular case of Hopf's classification theorem, which states that for a compact space of dimension at most n, there is a one-to-one correspondence between homotopy classes of maps to the n-sphere and elements of the nth Čech cohomology group. It can be found as Theorem 2.2 (p. 17) in [2], reformulated in [START_REF] Hu | Mappings of a normal space into an absolute neighborhood retract[END_REF].

Theorem 5.3.4. (Hopf classification theorem) Let X be a compact space of dimension ≤ n. There is a bijection between [X; S n ] and Ȟn (K; Z).

Application to Strong Computable Type

In this section, we provide a demonstration of our framework by revisiting previous results that establish the computable type property for pairs. We demonstrate that these results can be explained by the minimality of the pairs with respect to a specific Σ 0 2 invariant. By employing a single Σ 0 2 invariant, we can encompass a wide range of sets or pairs, thereby unifying the computabilitytheoretic aspect of the argument. This allows us to focus on a purely topological analysis for each specific set or pair, benefiting from the well-established field of topology. Consequently, our study of (strong) computable type becomes more explicitly connected to topology. Remarkably, most of the arguments presented in this section are primarily topological in nature.

Examples of E n -Minimal Pairs

In this section we show how E n -minimality covers many of the known examples of pairs having computable type. We also give a new example to illustrate its scope.

1) Chainable Continuum Between Two Points

Iljazović proved in [START_REF] Iljazović | Chainable and circularly chainable co-r.e. sets in computable metric spaces[END_REF] that if X is a continuum (a connected compact metric space) that is chainable from a point a ∈ X to a point b ∈ X, then the pair (X, {a, b}) has computable type. This result and its proof can be reformulated as the E 0 -minimality of the pair. We recall some definitions from [START_REF] Iljazović | Chainable and circularly chainable co-r.e. sets in computable metric spaces[END_REF].

Definition 5.4.1. Let X be a metric space. A finite sequence C of nonempty open subsets C 0 , . . . , C m of X is said to be a chain if

C i ∩ C j = ∅ ⇔ |i -j| ≤ 1, ∀i, j ∈ {0, . . . , m}. A chain C = {C 0 , . . . , C m } covers X if X ⊆ i C i ; for > 0, it is an -chain if max i (diam(C i )) < .
For a, b ∈ X, X is chainable from a to b if for every > 0, there exists an -chain C = {C 0 , . . . , C m } in X which covers X and such that a ∈ C 0 and b ∈ C m . Iljazović proved in [START_REF] Iljazović | Chainable and circularly chainable co-r.e. sets in computable metric spaces[END_REF] that if X is chainable from a to b, then the pair (X, {a, b}) has computable type. This result can be obtained using the invariant E 0 .

If X is compact connected and is chainable from a to b, then the pair (X, {a, b}) is E 0 -minimal.

Proof. Take S 0 = {a, b} and f = id S 0 . As X is connected, the function f has no continuous extension g : X → S 0 , because g -1 (a) and g -1 (b) would deconnect X. Hence, (X, {a, b}) is in E 0 .

Let x ∈ X \ {a, b} and > 0 be such that B(x, ) ⊆ X \ {a, b}. Let f : {a, b} → S 0 . The points a, b must be disconnected in X \ B(x, ). Indeed, let C = {C 0 , . . . , C m } be an -chain covering X with a ∈ C 0 and b ∈ C m . Let k be such that The disconnection can be turned into a continuous extension g : X \ B(x, ) → S 0 , sending U a to f (a) and U b to f (b). Therefore, (X \B(x, ), {a, b}) / ∈ E 0 . It implies that (X, {a, b}) is E 0 -minimal because any proper subset of X is contained in such a X \ B(x, ).

x ∈ C k . As diam(C k ) < , a / ∈ C k , b / ∈ C k and C k ⊆ B(x, ). Therefore X \ B(x,
We will cover other examples from [START_REF] Iljazović | Chainable and circularly chainable co-r.e. sets in computable metric spaces[END_REF] in Section 5.4.3, using another Σ 0 2 invariant.

2) Pseudo n-Cubes

In [START_REF] Horvat | Computability of pseudo-cubes[END_REF], the authors introduce the notion of a pseudo-n-cube, which is a compact pair (X, A) which can somehow be approximated by n-dimensional cubes and their boundaries. They prove that pseudo-n-cubes have computable type.

We show how the result can be proved using the invariant E n-1 , by showing how the E n-1minimality of the n-dimensional cube implies the E n-1 -minimality of pseudo-n-cubes.

We first recall the definition of pseudo-n-cubes from [START_REF] Horvat | Computability of pseudo-cubes[END_REF]. Let n ∈ N * and let S 0 1 , . . . , S 0 n and S 1 1 , . . . , S 1 n be the faces of the n-cube I n defined by

S 0 i = {(x 1 , . . . , x n ) ∈ I n : x i = 0}, S 1 i = {(x 1 , . . . , x n ) ∈ I n : x i = 1},
where i ∈ {1, . . . , n}. Let S = i S 0 i ∪ S 1 i be the boundary of I n .

Definition 5.4.2. A compact pair (X, A) in R n is called a pseudo-n-cube with respect to some finite sequence of compact subsets A 0 i and

A 1 i , 1 ≤ i ≤ n, if A = i A 0 i ∪ A 1 i , A 0 i ∩ A 1 i = ∅
and for each > 0 there exists a continuous injection f :

I n → R n such that 1. f (I n ) ⊆ X, 2. f (int R n (I n )) ∩ A = ∅, 3. X \ f (int R n (I n )) ⊆ N (f (S)), 4. d H (f (S 0 i ), A 0 i ) < and d H (f (S 1 i ), A 1 i ) < .
We now show how the proof that pseudo-n-cubes have computable type can be simplified by showing that they are E n-1 -minimal.

Proposition 5.4.2

Every pseudo-n-cube is E n-1 -minimal.

Proof. A pseudo-cube is approximated by cubes. We use Propositions 5.3.4 and 5.3.2 to derive E n-1minimality of pseudo-n-cubes from E n-1 -minimality of n-cubes (Proposition 5.3.6).

We recall that the pair (I n , S) ∼ = (B n , S n-1 ) is in E n-1 because the identity on S has no continuous extension to I n . A similar function h : A → S can be defined that has no extension to X.

The function h satisfies h(A 0 i ) ⊆ S 0 i and h(A 1 i ) ⊆ S 1 i and is built as follows. On an intersection A b 1 1 ∩ . . . ∩ A bn n (where b i ∈ {0, 1}), h sends every point to the vertex (b 1 , . . . , b n ) of I n . We then progressively extend h, sending intersections of n -1 sets to segments, intersections of n -2 sets to squares, then cubes, etc. and finally sending single sets A 0 i and A 1 i to the n -1-cubes which are the faces S 0 i and S 1 i of I n . All the extensions can be done using the Tietze extension theorem. Once h : A → S is defined, it has a continuous extension h to a neighborhood of A because S is an ANR. For sufficiently small and a cube (I n , S) embedded in X as in the definition, h is defined on S and sends S b i close to S b i , in particular disjoint from S 1-b i . Thus, h| S : S → S is homotopic to id S . In other words, the functions id S on the copies of (I n , S) are asymptotically homotopic to h : A → S. As id S has no continuous extension to I n , h has no continuous extension to X by Proposition 5.3.4. As a result, (X, A) ∈ E n-1 .

We now prove minimality. By definition of pseudo-n-cubes, for each there exists a copy (X i , A i ) of the n-cube in X satisfying the conditions of Proposition 5.3.2:

X i \ A i is open in X (indeed, it is the image by a continuous injection map of an open subset of R n in X ⊆ R n , so it is open by invariance of domain) and contained in X \ A, X \ (X i \ A i ) ⊆ N (A). If U ⊆ X \ A
is an open set intersecting X, then the same conditions are satisfied by the pairs (X i \ U, A i \ U ) and (X \ U, A). For sufficiently large i, U intersects X i so (

X i \ U, A i \ U ) / ∈ E n-1 by E n-1 - minimality of (X i , A i ). We can apply Proposition 5.3.2, which implies that (X \ U, A) / ∈ E n-1 . Therefore, (X, A) is E n-1
-minimal (we again use Corollary 5.3.2 which can be applied, as A ⊆ R n has empty interior, therefore has dimension at most n -1 by Theorem 3.7.1 in [60]).

3) A New Example

In this section, we showcase the application of E n -minimality in generating new instances of spaces with the computable type property, requiring minimal effort. We present a specific example to illustrate this approach, although numerous variations can be explored. Definition 5.4.3. The topologist's sine curve T , or Warsaw sine curve, is defined by

T = x, sin 1 x : 0 < x ≤ 1 ∪ {(0, y) : -1 ≤ y ≤ 1} .
We define the Warsaw saucer S by

S = (x, y, z) ∈ R 3 : 0 < x 2 + y 2 ≤ 1 and z = sin 1 √ x 2 +y 2 ∪ {(0, 0, z) : -1 ≤ z ≤ 1} .
We define its boundary ∂S as its bounding circle

∂S = (x, y, sin(1)) ∈ R 3 : x 2 + y 2 = 1 ∼ = S 1 .
The Warsaw saucer is obtained from the closed topologist's sine curve, making it rotate around the vertical segment, see Figure 5.2.

The set S is compact and connected but neither locally connected nor path-connected. It is neither a manifold, nor a simplicial complex, nor a pseudo-n-cube so none of the previous results applies to (S, ∂S). However, it can easily be proved to have strong computable type, because it is closely related to the pair (B 2 , S 1 ) and is therefore E 1 -minimal. The pair (S, ∂S) is E 1 -minimal and hence has strong computable type.

Proof. We show the E 1 -minimality of (B 2 , S 1 ) implies the E 1 -minimality of (S, ∂S).

The surface S can be approximated by copies of the disk B 2 that share the same boundary S 1 . Indeed, let

X i = (x, y, z) : 1 iπ ≤ x 2 + y 2 ≤ 1 and z = sin 1 √ x 2 +y 2 ∪ (x, y, 0) : x 2 + y 2 ≤ 1 iπ .
The sets X i converge to S, and they all have the same boundary S 1 . We know that (X i , S 1 ) = (B 2 , S 1 ) ∈ E 1 , moreover the same function, which is the identity id : S 1 → S 1 , has no continuous extension to X i . Therefore, this function has no continuous extension to S by Corollary 5.3.1.

We now prove E 1 -minimality. The projection π on the horizontal plane sends S to B 2 and is the identity on

∂S = S 1 . If X is a proper compact subset of S containing ∂S, then π(X) is a proper compact subset of B 2 . As (B 2 , S 1 ) is E 1 -minimal, (π(X), S 1 ) / ∈ E 1 so (X, ∂S) / ∈ E 1 by Proposition 5.3.1 (more directly, every f : S 1 → S 1 has a continuous extension F : π(X) → S 1 , inducing a continuous extension F = F • π : X → ∂S of f to X). As a result, (S, ∂S) is E 1 -minimal.

Compact n-Manifolds are H n -Minimal

It was proved in [START_REF] Iljazovic | Compact manifolds with computable boundaries[END_REF] and [START_REF] Iljazović | Semicomputable manifolds in computable topological spaces[END_REF] that compact manifolds with or without boundary have computable type, as pairs and single spaces respectively. In this section, we show how this result can be derived from classical results in algebraic topology implying that n-dimensional manifolds are H n -minimal, and applying Theorem 4.3.2.

The next result seems to be folklore but is not stated in any standard textbook on algebraic topology. However, it can be derived from classical results about homology of manifolds.

Theorem 5.4.1. Every connected compact n-manifold M is H n -minimal. Every connected compact n-manifold with boundary (M, ∂M ) is H n -minimal.
We give here the proof for manifolds without boundary, and put the proof for manifolds with boundary in the appendix (it is similar but requires more technical results about pairs). Proof. The result is a consequence of Theorem 5.3.3, which reduces H n to homology groups, and of classical results about homology groups of manifolds.

Let us use the abelian group G = R/Z. As it contains an element of order 2, namely 1/2, Corollary 7.12 in [START_REF] Bredon | Topology and Geometry[END_REF] implies that if M is a connected n-manifold, then H n (M ; G) = 0 if and only if M is compact. Therefore, if M is a compact connected n-manifold and x ∈ M , then

H n (M ; G) 0, H n (M \ {x}; G) ∼ = 0, because M \ {x} is a non-compact connected manifold.
If x ∈ M and B is an open Euclidean ball around x, then M \{x} deformation retracts to M \B, which means that there is a retraction r :

M \ {x} → M \ B such that if i : M \ B → M \ {x} is the inclusion map, then i • r is homotopic to id M \{x} . It is a classical result that deformation retractions preserve homology groups, so H n (M \ B) ∼ = H n (M \ {x}) ∼ = 0. Therefore, Theorem 5.3.3 implies that M ∈ H n and M \ B / ∈ H n . It implies that M is H n - minimal by Lemma 5.3.2: if X is a proper subset of M , then X ⊆ M \ B for some B, so X / ∈ H n .
Theorem 5.4.1 implies the result in [START_REF] Iljazovic | Compact manifolds with computable boundaries[END_REF] that compact manifolds with or without boundary have (strong) computable type (because a disconnected compact manifold is a finite union of connected compact manifolds, although they are not H n -minimal). However the proof is inherently different and has new implications. For instance, it implies that SCT M,∂M ≤ W C N by Theorem 4.4.2. Moreover, it immediately implies that other pairs have strong computable type, as follows.

Corollary 5.4.1. If M is a compact connected n-manifold with possibly empty boundary ∂M , then the pair

C(M, ∂M ) = (C(M ), M ∪ C(∂M ))
is E n -minimal hence has strong computable type.

Proof. All the assumptions of Proposition 5.3.9 are met: ∂M has empty interior in M , dim(M ) = n and dim(∂M

) = n -1. As (M, ∂M ) is H n -minimal by Theorem 5.4.1, C(M, ∂M ) is E n -minimal.
The cone of a manifold M is a manifold only when M is a sphere of a ball, so this result is indeed new.

We give another proof of this result in Section 6.7 using cycles, see Corollary 6.7.6.

Circularly Chainable Continuum Which Are Not Chainable

In [START_REF] Iljazović | Chainable and circularly chainable co-r.e. sets in computable metric spaces[END_REF], it is proved that every compact metrizable space which is circularly chainable but not chainable has computable type. The simplest examples of such sets are given by the circle and the Warsaw circle. We briefly show how the proof of this result can be reformulated in our framework by finding a suitable Σ 0 2 invariant. We already saw the notion of chain in Definition 5.4.1. We recall other related notions, taken from [START_REF] Iljazović | Chainable and circularly chainable co-r.e. sets in computable metric spaces[END_REF]. Let (X, d) be a metric space. A finite sequence C = (C 0 , . . . , C n ) of non-empty open subsets of X is called:

A chain if for all i, j ∈ {0, . . . , n}, C i ∩ C j = ∅ ⇐⇒ |i -j| ≤ 1, A circular chain if for all i, j ∈ {0, . . . , n}, C i ∩ C j = ∅ ⇐⇒ (|i -j| ≤ 1 or {i, j} = {0, n}). A quasi-chain if for all i, j ∈ {0, . . . , n}, |i -j| > 1 =⇒ C i ∩ C j = ∅.
Its mesh is defined by mesh

(C) = max 0≤i≤n (diam(C i )). It covers a set S ⊆ X if S ⊆ i C i .
A set S ⊆ X is chainable (resp. circularly chainable, quasi-chainable) if for every > 0 there exists a chain (resp. circular chain, quasi-chain) of mesh < covering S.

If S is connected, then S is chainable if and only if S is quasi-chainable (Lemma 30 in [START_REF] Iljazović | Chainable and circularly chainable co-r.e. sets in computable metric spaces[END_REF]).

Proposition 5.4.4

Not being quasi-chainable is a Σ 0 2 invariant in τ upV . If a compact space is circularly chainable but not chainable, then it is minimal satisfying this invariant and hence it has strong computable type.

Proof. If U ⊆ Q is a finite union of rational open balls, then we write U for the corresponding union of closed balls. By a standard compactness argument, a compact set S ⊆ Q is quasi-chainable if for every rational > 0 there exists a quasi-chain C = (C 0 , . . . , C n ) of mesh < covering S, with the following additional properties: C i ∩ C j = ∅ for |i -j| > 1, and diam(C i ) < . Finite unions of rational closed balls are effectively compact, so these additional properties are c.e., and whether C covers S is effectively open in the upper Vietoris topology. Therefore, being quasi-chainable is a Π 0 2 invariant.

If S is not chainable then S is not quasi-chainable. If in addition S is circularly chainable, then every proper compact subset of S is quasi-chainable. Indeed, let T S be compact, x ∈ S \ T and < d Q (x, T ), and let C = (C 0 , . . . , C n ) be a circular chain of mesh < covering S. We can assume that x ∈ C n , applying a circular permutation if needed. As diam(C n ) < < d Q (x, T ), the set C n is disjoint from T . As a result, (C 0 , . . . , C n-1 ) is a quasi-chain covering T .

It can be shown that if a compact space is in H 1 , then it is not quasi-chainable. We do not know whether the converse implication holds, in particular whether spaces that are circularly chainable but not chainable are H 1 -minimal.

A space that properly contains copies of itself

Theorem 4.3.2 states in particular that if a compact space X is minimal for some Σ 0 2 topological invariant then X has strong computable type. We show here that the converse implication does not hold. We actually prove more: we build a compact space X that has strong computable type and contains a proper subset X that is homeomorphic to X. It implies that X cannot be minimal for any topological invariant, because any invariant satisfied by X is also satisfied by X . Moreover, the space X is to our knowledge the first example of an infinite-dimensional space having strong computable type. The space X is built as follows. Let f : S 1 → Q be a so-called "space-filling curve", i.e. a surjective continuous function (such a function from [0, 1] to Q exists by the Hahn-Mazurkiewicz theorem, see Theorem 6.3.14 in [START_REF] Engelking | General topology[END_REF], and can be extended to the circle by joining the two endpoints). Let X = Q ∪ f B 2 be the adjunction space, which is the quotient of the disjoint union Q B 2 by the equivalence relation generated by x ∼ f (x) for x ∈ S 1 = ∂B 2 . Note that X contains a proper subset which is homeomorphic to X: indeed, X properly contains Q, which contains a copy of X. Proposition 5.4.5

If f : S 1 → Q is surjective continuous, then the space Q ∪ f B 2 has strong computable type.
Proof. Although X cannot be H 2 -minimal, we show that there is a continuous function q : X → S 2 which is not null-homotopic (i.e. X ∈ H 2 ) and such that for any compact subset Y X, the restriction q| Y : Y → S 2 is null-homotopic. The condition that f is surjective implies that Q has empty interior in X: every y ∈ Q is the image by f of some x ∈ ∂B 2 , which is a limit of a sequence x n ∈ B 2 \∂B 2 ; therefore in X, y is the limit of x n ∈ X \Q. The quotient space X/Q is homeomorphic to B 2 /∂S 2 , which is homeomorphic to S 2 , so we can type the quotient map q : X → X/Q as q : X → S 2 . Proposition 0.17 in [START_REF] Hatcher | Algebraic Topology[END_REF] states that if a pair (X, A) satisfies the homotopy extension property and A is contractible, then the quotient map q : X → X/A is a homotopy equivalence. The pair (X, Q) satisfies the homotopy extension property because X is obtained by attaching a disk to Q along its boundary (see the proof of Proposition 0.16 in [START_REF] Hatcher | Algebraic Topology[END_REF]). The space Q is contractible, so the quotient map q : X → X/Q ∼ = S 2 is a homotopy equivalence, i.e. there exists a function p : S 2 → X such that p • q is homotopic to id X and q • p is homotopic to id S 2 . It implies that q is not null-homotopic: otherwise the function id S 2 , which is homotopic to q • p, would be null-homotopic as well, i.e. S 2 would be contractible. We now show that if Y is a compact proper subset of X, then the restriction q| Y : Y → S 2 is null-homotopic. As Q has empty interior in X, the interior of B 2 is dense in X so it contains a point x ∈ X \ Y . The space X \ {x} is contractible, because it deformation retracts to Q which is contractible. Therefore, the restriction of q to X \ {x} is null-homotopic, so its restriction to Y ⊆ X \{x} is null-homotopic, by restriction of the homotopy. As a result, for a compact set Y ⊆ X, one has Y = X if and only if the restriction of q to Y is null-homotopic. We now prove that X has computable type. Let X 0 ⊆ Q be a semicomputable copy of X, and φ : X 0 → X a homeomorphism. The previous discussion implies that for an open set U ⊆ Q, U intersects X 0 if and only if the restriction of q • φ to X 0 \ U is null-homotopic. We now use the properties of the numberings ν [X;Y ] from Theorem 5.2.1. Let i 0 be an index of a constant computable function

f i 0 : Q → S 2 . Note that i 0 ∈ dom(ν [Z;S 2 ] ) for all Z ⊆ Q. Let i 1 be such that ν [X 0 ;S 2 ] (i 1 ) is the homotopy class of q • φ. The definition of ν [X;Y ] implies that for any open set U ⊆ Q, i 1 ∈ dom(ν [X 0 \U ;S 2 ] ) and ν [X 0 \U ;S 2 ] (i 1 ) is the homotopy class of the restriction of q to X 0 \ U . We saw that a basic open set U ⊆ Q intersects X 0 if and only if q| X 0 \U is null-homotopic, which is equivalent to ν [X 0 \U ;S 2 ] (i 1 ) = ν [X 0 \U ;S 2 ] (i 0 )
. As X 0 is semicomputable, so is X 0 \ U , therefore this predicate is c.e. and X 0 is computable, which shows that X 0 has computable type. The argument holds relative to any oracle, i.e. if X 0 is semicomputable relative to an oracle O, then X 0 is computable relative to O. Therefore, X has strong computable type.

Π 0 1 Invariants

In this section, we give a complete characterization of the topological invariants in the lowest descriptive complexity class, namely Π 0 1 . We show that this level of complexity can only detect connectedness properties of the space. Definition 5.5.1. Let 0 ≤ p ≤ n be natural numbers. The topological invariant C n,p is defined by: X ∈ C n,p iff X has at most n connected components, among which at most p non-trivial ones.

As particular cases,

X ∈ C n,n ⇐⇒ X has at most n connected components, X ∈ C n,0 ⇐⇒ X contains at most n points. These invariants are examples of Π 0 1 invariants. Proposition 5.5.1 Let 0 ≤ p ≤ n. The invariant C n,p is Π 0 1 .
Proof. We show that X / ∈ C n,p iff there exists n + 1 disjoint open sets covering X, each one intersecting X, or p + 1 disjoint open sets covering X, each one intersecting X in at least n -p + 1 distinct points. This property is Σ 0 1 . Of course, if X / ∈ C n,p then one of these coverings exists: either X has at least n + 1 connected components, or X has at least p + 1 non-trivial connected components, which then contain as many points as needed. Conversely, if the first type of covering exists, then X has more than n components so X / ∈ C n,p . Now assume that the second type of covering exists. If X has more than p non-trivial components, then X / ∈ C n,p . If X has at most p non-trivial components, then one of the open sets does not contain an infinite component, so it contains at least n -p + 1 singletons. In total, X contains at least p + (n -p + 1) = n + 1 components, so X / ∈ C n,p .

Moreover, these invariants generate all the Π 0 1 invariants.

Theorem 5.5.1. Let P be a non-trivial topological invariant. The following statements are equivalent:

1. P ∈ Π 0 1 , 2. P ∈ Π 0 1 , 3. P is a finite union of C n,p 's.
Observe that 3. ⇒ 2. follows from Proposition 5.5.1 and 2. ⇒ 1. is obvious. The next section is devoted to the proof of 1. ⇒ 3.

Approximations

We introduce the notion of approximation, which is the key ingredient underlying the proof. Definition 5.5.2. A compact space X approximates Y if some copy Y 0 ⊆ Q of Y is a limit of copies of X (in the Vietoris topology, or equivalently the Hausdorff metric). Proposition 5.5.2

Let X, Y be compact spaces. The following statements are equivalent:

X approximates Y , Every Π 0 1 invariant satisfied by X is satisfied by Y .
Proof. If X approximates Y and X satisfies a Π 0 1 invariant I, then all the copies of X belong to I and so do their limits.

Conversely, assume that every Π 0 1 invariant satisfied by X is satisfied by Y . Let I be the set of limits of copies of X. By definition, I is closed, i.e. Π 0 1 . I is a topological invariant: if K ⊆ Q is a limit of copies X n of X and K ⊆ Q is homeomorphic to K, then K is a limit of copies of X, because the homeomorphism f : K → K can be extended to a homeomorphism F : Q → Q (Theorem 5.2.4 in [START_REF] Van Mill | The Infinite-Dimensional Topology of Function Spaces[END_REF]), and the copies F (X n ) of X converge to F (K) = K . Strictly speaking, Theorem 5.2.4 in [START_REF] Van Mill | The Infinite-Dimensional Topology of Function Spaces[END_REF] can be applied only if K, K ⊆ (0, 1) N . If it is not the case, we apply the argument to copies of K, K scaled down so that they fit in (0, 1) N , and observe that K is a limit of such copies, taking scaling factors arbitrarily close to 1, which are limits of copies of X.

Finally, I is satisfied by X hence by Y , i.e. some (actually, every) copy of Y is a limit of copies of X.

The proof also shows that in the definition of approximation, one could equivalently replace "some copy" by "every copy".

The next result gives a particularly simple criterion to establish that a space approximates another space. Proposition 5.5.3

Let X, Y be compact spaces. If there exists a continuous surjective function f : X → Y , then X approximates Y .

Proof. We assume that X, Y are embedded in Q. The function f : X → Y is a limit of injective continuous functions f n : X → Q, defined as follows. f n (x) consists of the first n coordinates of f (x), followed by x (note that x and f (x), which are points of Q, are sequences of real numbers, therefore it makes sense to insert the first n terms of f (x) at the beginning of x). f n is injective, continuous, and is closer and closer to f as n grows. Therefore, Y is the limit of the sets f n (X), which are copies of X.

Here are some properties of approximation related with the disjoint union (see Definition 3.1.2) and connectedness. Proposition 5.5.4

1. The approximation relation is a pre-order, i.e. it is reflexive and transitive, (2) Given two sets A 0 , A 1 ⊆ Q, their disjoint union A B can be realized as ({0}

2. If X i approximates Y i for i ∈ {0, 1}, then X 0 X 1 approximates Y 0 Y 1 , 3. If X = ∅, then X Y approximates X,
× A 0 ) ∪ ({1} ∪ A 1 ) ⊆ Q. Let Y 0 , Y 1 ⊆ Q be limits of copies X n 0 , X n 1 of X 0 , X 1 respectively. The realization of Y 0 Y 1 is the limit of the realizations of X n 0 X n 1 , which are copies of X 0 X 1 . (3) Let x 0 ∈ X. X is the image of X Y under a continuous function f : X Y → X, defined by f (x) = x for x ∈ X and f (y) = x 0 for y ∈ Y . ( 4
)
As X is non-trivial, it contains two distinct points, which differ in at least one coordinate. The projection along that coordinate is a continuous function sending X to a line segment, so X approximates the line segment. The line segment approximates every finite connected graph, because a such a graph is a continuous image of the line segment (a path visiting all the edges, possibly with repetitions, is nothing else than a continuous surjective function from the line segment to the graph). It remains to show that every connected compact space Y is a limit of finite connected graphs.

Given > 0, let V ⊆ Y be a finite set such that every point of Y is -close to a point of V . Consider the graph G with vertex set V , with an edge between u and v if d(u, v) < 2 . G is naturally embedded in Q, and each point of an edge is -close to one of the endpoints, so d H (G, Y ) < . G must be connected: if there is a non-trivial partition V = V 1 V 2 with no edge between V 1 and V 2 , then the open -neighborhoods of V 1 and V 2 cover Y and do not intersect, so they disconnect Y , giving a contradiction.

We now have all the ingredients to prove the main result of this section. Proof of Theorem 5.5.1. Let P be a non-trivial Π 0 1 invariant. We first show that there is an upper bound on the number of connected components of spaces in P. We assume the contrary and show that every space belongs to P, contradicting the non-triviality of P. First, a space X with at least n components approximates the finite set F with n points, because there is a continuous surjective function f : X → F . As the finite sets are dense in K(Q), if the closed set P contains every finite set, it contains every compact set. Therefore, P is trivial, giving the desired contradiction.

Let X ∈ P have n connected components, p of them being non-trivial. X approximates every set Y ∈ C n,p : to each connected component Y i of Y one can associate a connected component X i of X in an injective way, such that if Y i is non-trivial then X i is non-trivial. Each Y i is approximated by X i by Proposition 5.5.4, item 4., so X approximates Y by Proposition 5.5.4, items 2. and 3..

Σ 0 2 Invariants

Our goal is to analyze the separation power of Σ 0 2 invariants. A natural invariant of that complexity is given by the topological dimension. Proposition 5.6.1

Having dimension at least n is a Σ 0 2 invariant in the Vietoris topology.

Proof. Let X be a compact subset of Q. Using standard compactness arguments, one can show that dim(X) ≥ n ⇐⇒ there exists a finite sequence U 0 . . . , U k of finite union of rational balls of Q covering X such that there is no finite sequence V 0 , . . . , V l of finite unions of rational balls covering X, such that each closure V i is contained in some U j and each intersection of n + 2 sets V i 's is empty. We now use the fact that Q is effectively compact, i.e. given a Σ 0 1 subset U of Q, one can semidecide whether U = Q (see [START_REF] Iljazović | Computability of subsets of metric spaces[END_REF] for instance); in other words, given a Π 0 1 subset of Q, one can semi-decide whether it is empty. Therefore, whether a closed ball is contained in an open set, and whether a closed set is empty are semi-decidable or Σ 0 1 conditions, so the whole formula is Σ 0 2 . We first introduce a refinement of the notion of approximation and prove an analog to Proposition 5.5.2.

Strong Approximation

Definition 5.6.1. Let X, Y be compact spaces. We say that X strongly approximates Y , written X Y , if there exists a copy X 0 ⊆ Q of X such that for every > 0, some copy of Y is a limit of -deformations of X 0 .

Note that the definition would be equivalent if one requires the -deformations of X 0 to be images of X 0 under injective -functions. Indeed, as in Proposition 5.5.3, an -function is a limit of injective -functions. Remark 5.6.1

As for the notion of approximation, if X strongly approximates Y , then the condition actually holds for every copy X 0 ⊆ Q of X.

Indeed, let > 0 and let h : Q → Q be a homeomorphism sending X 0 to X 0 . By uniform continuity take δ > 0 such that d

Q (y 1 , y 2 ) < δ implies d Q (h(y 1 ), h(y 2 )) < /2 and suppose Y = lim n→∞ f n (X 0 ) where f n s are δ-functions. We have Y = lim n→∞ f n • h -1 (X 0 ) hence h(Y ) = lim n→∞ h • f n • h -1 (X 0 ) as h is continuous. In addition, d X 0 (h • f n • h -1 , id X 0 ) < : let x ∈ X 0 and y = h -1 (x), we have d Q (h • f n • h -1 (x), x) = d Q (h(f n (h -1 (x)), h(h -1 (x))) = d Q (h(f n (y)), h(y)), but d Q (f n (y), h(y)) < δ so d Q (h(f n (y)), h(y)) < /2.
Hence, the condition actually holds for the arbitrarily copy X 0 ⊆ Q of X.

Theorem 5.6.1. Let X, Y be compact spaces. The following statements are equivalent: X strongly approximates Y , Every Σ 0 2 invariant satisfied by X is satisfied by Y .

Proof. Assume that every Σ 0 2 invariant satisfied by X is satisfied by Y . Let X 0 ⊆ Q be a copy of X and let > 0. Consider the following topological invariant P: Y ∈ P iff there exists δ < and a copy Y 0 of Y which is a limit of δ-deformations of X 0 . We show that it is Σ 0 2 . Let P δ be the set of limits of δ-deformations of X 0 . It is closed, i.e. Π 0 1 . Let f i : Q → Q be a dense sequence of computable injective functions. It is not difficult to see that Y ∈ P iff there exists δ < and i such that f i (Y ) ∈ P δ , which is a Σ 0 2 condition. Now, X obviously satisfies P, so Y satisfies P as well by assumption. As it is true for every > 0, X strongly approximates Y . Now assume that X strongly approximates Y and let P be a Σ 0 2 invariant satisfied by X. One has P = n∈N P n where each P n is Π 0 1 . For every homeomorphism f : Q → Q, f (X) ∈ P so f (X) ∈ P n for some n. Therefore, there exists n ∈ N such that for "many" f , f (X) ∈ P n . It can be made precise by using Baire category on the Polish space of homeomorphisms from Q to itself. It implies that there exists n ∈ N, f : Q → Q and > 0 such that for every g : Q → Q satisfying d(f, g) < , g(X) ∈ P n . As a result, every -deformation of f (X) belongs to P n . As P n is closed, every limit of -deformation of X belongs to P n , so some copy of Y is in P n ⊆ P. Therefore, Y satisfies P.

Theorem 5.6.1 gives a very effective way of proving that there is no Σ 0 2 invariant separating X from Y , i.e. satisfied by X but not by Y : one simply needs to design -deformations of X converging to a copy of Y , for every .

The Closed Topologist's Sine Curve and the Line Segment

We illustrate this technique by showing that the line segment and the closed topologist's sine curve cannot be separated by Σ 0 2 invariants.

Definition 5.6.2. The closed topologist's sine curve is the set

S = {(x, sin(1/x)) : x ∈ (0, 1]} ∪ {(0, y) : y ∈ [-1, 1]}.
Proposition 5.6.2

The closed topologist's sine curve and the line segment cannot be distinguished by Σ 0 2 invariants.

Proof. Using Theorem 5.6.1, we simply have to show that S and I = {(x, 0) : x ∈ [0, 1]} strongly approximate each other. We work in the plane R 2 , which embeds in Q.

Let > 0, we define continuous functions f n : I → R 2 as follows:

f n (x, 0) = (x, sin( 1 x )) if x ≥ 1 nπ , (x, 0) if x ≤ 1 nπ .
Each f n is an -function, and the sets f n (I) converge to a copy of S, vertically scaled by . Therefore, I strongly approximates S. First observe that when embedding a pointed space (Z, z 0 ) in Q, one can always send z 0 to 0 ∈ Q because Q is homogeneous, i.e. every point of Q can be sent to any other point by some homeomorphism (Theorem 1.6.6 in [START_REF] Van Mill | The Infinite-Dimensional Topology of Function Spaces[END_REF]).

Let

Q 0 = {x ∈ Q : ∀i, x 2i = 0} and Q 1 = {x ∈ Q : ∀i, x 2i+1 = 0}
. These subspaces of Q are homeomorphic to Q. By the previous observation, we can embed X in Q 0 and Y in Q 1 , so that x 0 = y 0 = 0.

As x 0 = 0 is not isolated in X, the function d(x, 0) takes infinitely many values in B(0, 1/n) ∩ X for any n. Therefore, the set

P n := {1 -nd(x, 0) : x ∈ B(0, 1/n) ∩ X}
is infinite, so some copy Y n of Y is a limit of copies of P n . In other words, there exist continuous functions h n : P n → Q 1 such that h n (P n ) converge to some copy Y of Y . We can moreover assume that h n (0) = 0.

Let > 0. We define g n : X → Q as follows:

g n (x) =      x if 2/n ≤ d(x, 0), (nd(x, 0) -1)x if 1/n ≤ d(x, 0) ≤ 2/n, h n (1 -nd(x, 0)) if d(x, 0) ≤ 1/n.
If n is sufficiently large, then one has d(g n (x), x) ≤ 2 for all x ∈ Q. When n → ∞, g n (X) converges to X ∨ Y .

Separating Finite Topological Graphs

In order to understand which spaces can be separated by Σ 0 2 invariants, we focus on a restricted class of spaces, namely the finite topological graphs. We obtain a characterization of when a graph strongly approximates another graph. As a consequence, we show that Σ 0 2 invariants can separate any two graphs that are not homeomorphic. However, the result is not symmetric: for instance, for each n ≥ 2 there is a Σ 0 2 invariant satisfied by the star with n + 1 branches but not by the star with n branches, but not vice-versa.

Modulus of Local Connectedness

Proving that a space X can be separated from a space Y by some Σ 0 2 invariant is not always easy, because only few classical invariants are Σ 0 2 , and we often need to design specific ones. Theorem 5.6.1 gives an alternative strategy, by showing that X does not strongly approximate Y . The proof of the theorem implicitly defines a Σ 0 2 invariant (namely, having a copy which is a limit of -deformations of X). However this approach has two limitations: It may not be simple to show that X does not strongly approximate Y , The argument is not effective: it yields a Σ 0 2 (boldface) invariant, but not a Σ 0 2 (lightface) one.

In this section, we introduce a quantitative measure of local connectedness of the space, which can be used to separate many spaces and is effective, i.e. computable in some weak sense. We will see in the next section that it induces a complete invariant for the class of finite connected graphs. Definition 5.7.1. Let (X, d) be a compact connected metric space. Its modulus of local connectedness is a function η X : (0, +∞) → [0, +∞) defined by η X (r) = min{s ∈ [0, +∞) : ∀x ∈ X, B(x, r) is contained in a connected component of B(x, s)}.

A compact subset of Q inherits the metric from Q. Let K conn (Q) be the space of connected compact subsets of Q. Proposition 5.7.1

Let X be compact connected. The function η X is well-defined and non-decreasing. Moreover, the set {(X, r, s) :

η X (r) > s} is Σ 0 1 in the product space K conn (Q) × R × R.
Proof. The set C X := {(r, s, x) : B(x, r) is contained in a connected component of B(x, s)} is Π 0 1 relative to X. Indeed, (r, s, x) / ∈ C X iff there exist disjoint open sets U, V both intersecting B(x, r) and such that B(x, s) ⊆ U ∪ V . By compactness of X, the equivalence still holds if one additionally requires U, V to be finite unions of rational open balls and their corresponding unions of closed balls are disjoint. The existence of such open sets can then be effectively tested, showing that the complement of C X is Σ 0 1 . For connected X with diameter d, the set defining η X (r) contains d. As this set is closed and lower bounded by 0, it has a minimum, showing that η X (r) is well-defined. It is obviously non-decreasing.

One has η X (r) > s iff ∃x ∈ X, (r, s, x) / ∈ C X which is Σ 0 1 as C X is Π 0 1 and one can semidecide whether X intersects a Σ 0 1 set. Observe that X is locally connected if and only if inf r>0 η X (r) = 0. It implies that being locally connected is Π 0 3 , as proved in [START_REF] Debs | The descriptive complexity of connectedness in Polish spaces[END_REF]. The key property of the modulus of local connectedness is that it does not increase underdeformations. Proposition 5.7.2

If Y is a limit of -deformations of X, then η Y (r) ≤ η X (r + 2 ) + .

Proof. We first show the inequality when Y is an -deformation of X. Let f : X → Y be a surjective -function. Let r > 0 and s < η Y (r). There exists x, y ∈ Y such that d(x, y) < r and x, y are not in the same connected component of B(x, s) ∩ Y . Let x = f -1 (x) and y = f -1 (y). One has d(x , y ) < r +2 , and x , y are not in the same connected component of B(x , s-)∩X. Indeed, if C ⊆ B(x , s -) ∩ X is connected and contains x , y , then f (C) ⊆ B(x, s) ∩ Y is connected and contains x, y, giving a contradiction. As a result, η X (r + 2 ) > s -. As this inequality holds for any s < η Y (r), we obtain the result.

As the function Y → η Y (r) is lower semicontinuous, the inequality is preserved by taking limits, so it holds for any Y which is a limit of -deformations of X.

Effective vs Non-Effective Classes

We now show that if X is a finite topological graph, then the Σ 0 2 invariants are no more expressive than the Σ 0 2 for separating X from other spaces. We use the modulus of local connectedness to define a family of Σ 0 2 invariants, which are as powerful as arbitrary Σ 0 2 invariants to separate X from other spaces. Theorem 5.7.1. Let X be a connected finite topological graph and Y a compact space. If there exists a Σ 0 2 invariant satisfied by X and not Y , then there exists a Σ 0 2 such invariant.

Proof. If Y is disconnected, then "being connected" is a Π 0 1 invariant satisfied by X but not by Y . Now assume that Y is connected.

We fix a copy X 0 of X in Q. For rational α > 0, we define an invariant P α as follows.

Y ∈ P α ⇐⇒ there exists β < α and a copy Y 0 of Y such that d H (X 0 , Y 0 ) < β and η Y 0 (r) ≤ η X 0 (r + 2β) + β for all r > 0.

Claim 5.7.1

If α is rational, then P α is a Σ 0 2 invariant.

Proof of the claim.

We just give an outline of the argument. There exists a computable sequence (f i ) i∈N of injective functions f i : Q → Q, which is dense in the space of injective continuous functions from Q to Q. The definition of P α would be equivalent if one replaces arbitrary copies Y 0 with copies f i (Y ). Therefore, the quantification over Y 0 can be replaced by a quantification over i ∈ N. There rest of the formula is made of Σ 0 1 and Π 0 1 statements, so the whole expression is Σ 0 2 . Note that X ∈ P α for every α > 0. We assume that Y ∈ P α for every α > 0, and prove that X strongly approximates Y , implying that every Σ 0 2 invariant satisfied by X is satisfied by Y , by Theorem 5.6.1.

Let > 0. We show that some copy of Y is a limit of -deformations of X 0 . The idea is that we can cut X into small line segments, and that each line segment -approximates a small connected subset of (a copy of) Y . Putting all the pieces together, X -approximates that copy of Y . Let us now give the details.

Let δ > 0 be small (more precisely, we want δ < /3 and η X 0 (3δ) < /12). We subdivide each edge of X by adding new vertices, so that the new edges have diameters < δ. Let V and E be the new sets of vertices and edges respectively. Let α > 0 be such that for each edge e ∈ E, its open α-neighborhood U e has diameter < δ (we also need α < δ and α < /12). Let Y 0 be a copy of Y witnessing that Y ∈ P α . As d H (X 0 , Y 0 ) < α, Y 0 is covered by the U e 's.

One has η Y 0 (δ) ≤ η X 0 (δ + 2α) + α ≤ η X 0 (3δ) + α < /6. Each Y 0 ∩ U e has diameter < δ, so it is contained in a connected set C e ⊆ Y 0 of diameter /3. As the U e 's cover Y , one has Y 0 = e C e .

For each vertex v ∈ V , we choose a point y v ∈ Y 0 such that d(v, y v ) < α. For each edge e that is incident to v, one has

y v ∈ Y 0 ∩ U e ⊆ C e .
For each edge e, C e is connected so it is approximated by the segment e. In other words, there exist continuous functions f e n : e → Q such that f e n (e) converge to C e when n grows. Moreover, we can make sure that if v is an endpoint of e, then f e n (v) = y v . Therefore, for each n, the functions f e n can be concatenated into one continuous function f n : X → Q, sending each v to y v . The sets f n (X) converge to Y when n grows. We finally show that for sufficiently large n, f n is an -function. Let e be an edge and v one of its endpoints. One has diam(e) < δ and diam(C e ) ≤ /3, so for x ∈ e and y ∈ C e ,

d(x, y) ≤ d(x, v) + d(v, y v ) + d(y v , y) < δ + α + /3 < .
If n is sufficiently large, then f e n (x) is close to C e , so d(x, f e n (x)) < .

The Σ 0 2 invariants defined in the proof are rather ad hoc and not intuitive that is why finding a more natural family of Σ 0 2 invariants achieving the same effect would be appreciated.

Separating Graphs

In this section, we prove that two non-homeomorphic finite graphs can be separated by some Σ 0 2 invariant. First, we show that a graph G can be separated from another graph H precisely when H cannot be contracted to G, even after subdivision. We allow graphs with multiple edges and loops. Topologically, it makes no difference because subdividing such a graph results in simple graph which is topologically the same. However, edge contractions can create multiple edges and loops.

In a graph, an edge contraction consists in removing an edge and identifying its two endpoints. A contraction is a sequence of edge contractions. A subdivision consists in adding new vertices inside the edges, and splitting the edges accordingly (it does not change the topology of the graph). We refer to [START_REF] Heggernes | Contracting graphs to paths and trees[END_REF] for details on contractions.

Theorem 5.7.2. Let G, H be finite connected graphs that are not singletons. The following statements are equivalent:

There is a Σ 0 2 invariant satisfied by G but not H,

No subdivision of H can be contracted to G.

Note that the same result holds for Σ 0 2 invariants, by Theorem 5.7.1. The proof of this result is presented in the end of the section 5.7.3.

Example 5.7.1

For instance, every Σ 0 2 invariant satisfied by the line segment is satisfied by the graphs that have a bridge, i.e. an edge whose removal disconnects the graph. Every Σ 0 2 invariant satisfied by the circle is satisfied by the graphs that have a cycle. Moreover, the line segment and the circle can be separated by some Σ 0 2 invariant, in both directions.

A consequence of this result is that non-homeomorphic graphs can be separated by some Σ 0 2 invariant, because they cannot be contracted to each other, even after subdivisions.

Theorem 5.7.3. If G, H are non-homeomorphic finite connected graphs, then there exists a Σ 0 2 invariant satisfied by one of them but not the other.

Proof. In order to apply Theorem 5.7.2, we need to show that G and H cannot be contracted to each other, even after subdivisions. We assume that some subdivision of G contracts to H and some subdivision of H contracts to G, and prove that G and H must be homeomorphic. One may suggest to extend Theorem 5.7.3 to larger families of spaces. A natural such family is given by the finite simplicial complexes, which are higher-dimensional generalizations of graphs. However, the analog statement fails. For instance, if X is the disk and Y is the wedge of two disks, then these spaces are 2-dimensional simplicial complexes that are not homeomorphic, however they cannot be separated by Σ 0 2 invariants (it is not difficult to show that they strongly approximate each other). It raises the following question: what is the level of complexity needed to separate 2-dimensional finite simplicial complexes? what about the n-dimensional ones?

Proof of Theorem 5.7.2

We present the proof of Theorem 5.7.2, which can be reformulated this way: if G, H are finite topological graphs, then one has G H iff some subdivision of H can be contracted to G.

We use the following characterization of contractibility from [START_REF] Heggernes | Contracting graphs to paths and trees[END_REF].

Lemma 5.7.1

G is a contraction of H iff there exists a surjective map ϕ :

V H → V G such that: Each ϕ -1 (s) is connected in G,
For every pair of distinct vertices s, t of G, (s, t) is an edge of G iff there exists an edge e = (u, v) of H such that ϕ(u) = s and ϕ(v) = t.

Proof of Theorem 5.7.2. One implication is easy: if G is obtained from H by an edge contraction, then we show that G strongly approximated H. Let e be the edge of H that is contracted and w be the vertex of G replacing e. Consider a small neighborhood of w, which is a star. For any > 0, one can -deform this star, so that the endpoints are fixed and a small region around w is collapsed to an edge. The result of this -deformation is a graph in which w has been replaced by an edge; in other words, it is a copy of H. Therefore, G strongly approximates H.

We now prove the other direction, which is much more involved. Let us first discuss why subviding H cannot be avoided. Consider the following example: H is just two vertices joined by an edge, and G is the subdivision of H obtained by adding a vertex in the midde. In that case, G and H are topologically the same, however H cannot be contracted to G, because it has less vertices. The best we can hope is that some subdivision of H can be contracted to G.

The strategy is as follows:

We fix a copy of G, a sufficiently small > 0 and a copy of H which is a limit of -deformations of G,

We subdivide H so that each edge of H is entirely contained in the -neighborhood of some edge of G,

On each edge e of G, we choose a point a e which is far from the vertices of both G and H,

We show that a e is close to exactly one edge of H, When removing the points a e from G, one is left with a disjoint union of stars centered at the vertices of G,

We show that H can be contracted to G, by sending each vertex v of H to the center of the star that is closest to v (and using Lemma 5.7.1).

We fix a copy of G and assume that the edges are straight and the distances between two distinct vertices is at least 1. One has η G (r) ≤ r. Let n be the number of vertices of H. Let e be an edge of G. There must be a point a in the middle third of e which is far from every vertex of H. Take n + 1 points regularly distributed on the middle third of e. They are all at distance at least 1 3n from each other. A vertex of H can be 1 6n -close to at most one of these points, so by the pigeonhole principle, some of these points is 1 6n -far from V H . We choose such a point for each edge e of G and call it a e .

Let δ ≤ 1 6n and < δ/6. Let G n be /2-deformations of G converging to a copy of H. We call this copy H for simplicity.

We now subdivide H, so that every edge of H is entirely close to an edge of G.

Claim 5.7.2

We can subdivide H so that each edge of H is entirely contained in the -neighborhood of some edge of G.

Proof. The -neighborhoods of the edges of G form an open cover of the compact set H. Let µ be a Lebesgue number of the cover. We subdivide each edge of H so that the new edges have diameters smaller than µ, and are therefore contained in the -neighborhood of some edge of G.

From now on, we assume that H has been subdivided.

Claim 5.7.3

For every a ∈ G, if d(a, V H ) > δ then B(a, ) intersects H in exactly one edge of H.

Proof. First, B(a, ) intersects H, because the Hausdorff distance between G and H is less than , and a ∈ G.

As H is a limit of -deformations of G, one has η H (r) ≤ η G (r + 2 ) + = r + 3 , so η H (2 ) ≤ 5 . If x and y are two points of B(a, ) ∩ H, then d(x, y) < 2 so x and y are connected in B(x, 5 ) ∩ H ⊆ B(a, δ) ∩ H. This set is disjoint from V H , so the only way for x and y to be connected in that set is that they belong to the same edge of H.

In particular, for each edge e of G, B(a e , ) intersects exactly one edge f e of H. Conversely, Claim 5.7.4

Each edge f of H can intersect at most one ball B(a e , ). In particular, the map e → f e is injective.

Proof of the claim. If e = e, then d(a e , e) ≥ 1/3, so B(a e , ) ∩ N (e) = ∅ (assuming ≤ 1/6). Because of the preliminary subdivision of H, each edge f of H is contained in N (e) for some edge e of G, therefore it does not intersect B(a e , ) for e = e, so it can only intersect B(a e , ).

It follows that the map e → f e is injective. Let e = e. As f e intersects B(a e , ), it cannot intersect B(a e , ). As f e intersects this ball, one has f e = f e .

Let e be an edge of G. As e is convex, the function r : Q → e sending a point x to the closest point on e is continuous. Its restriction to f e is a continuous function h : f e → e. The edge e is a copy of the unit interval [0, 1] and therefore inherits its natural ordering ≤ (there are two possible orientations, we choose an arbitrary one).

Let u, v be the endpoints of f e .

Claim 5.7.5

On the edge e, a is between h(u) and h(v). We use Lemma 5.7.1 and define a map ϕ : V H → V G sending v ∈ H to the center of the star of y v .

Let f = (u, v) be an edge of H such that ϕ(u) = ϕ(v). f is -close to an edge e = (s, t) of G. e is 1/3-far from every star centered at a vertex other than s and t, so f is (1/3 -)-far from these stars. As u and v are -close to the stars of ϕ(u) and ϕ(v) respectively, one must have ϕ(u) = s and ϕ(v) = t (or the symmetric case). As a result, (ϕ(u), ϕ(v)) is the arc e of G.

If e = (s, t) is an edge of G, then f e = (u, v) satisfies ϕ(u) = s and ϕ(v) = t because u, v stand on opposite sides of a e . Claim 5.7.6

Each ϕ -1 (s) is connected.

Proof of the claim.

Let u, v ∈ ϕ -1 (s). y u and y v belong to the star S s centered at s. The star is connected, so there exists a sequence of points y 0 , y 1 , . . . , y k ∈ S s with y 0 = y u , y k = y v and d(y i , y i+1 ) < . As G and H are /2-close in the Hausdorff metric, there exist x 0 , . . . , x k ∈ H such that d(x i , y i ) ≤ /2. One has d(x i , x i+1 ) < 2 , so x i and x i+1 are connected in B(x i , 5 ) ∩ H. Therefore, u and v are connected in N 5 (S s ) ∩ H, i.e. there is a path from u to v in H contained in N 5 (S s ). All the vertices of this path are close to S s , so they belong to ϕ -1 (s). As a result, u and v are connected in ϕ -1 (s).

Conclusion

T he findings regarding strong computable type presented in the preceding chapter serve as a catalyst for delving into the descriptive complexity of topological invariants, a subject that this chapter explores in depth. Namely, we proved that ANRs allow us to define invariants of low effective descriptive complexity; using this result, we defined two Σ 0 2 invariants thanks to which we revisited several results about computable type from the literature and obtained new ones. In addition, we characterized the Π 0 1 invariants using connectedness. We studied the expressive power of Σ 0 2 invariants and proved that the effective versions (Σ 0 2 invariants) can separate between finite topological graphs. The objective of this chapter is to examine the surjection and -surjection properties, offering methods to confirm or disprove them, particularly through the utilization of homotopy and homology theories. Additionally, this chapter aims to present various applications of these techniques. The obtained results are applicable to finite simplicial complexes, as well as slightly broader classes of compact metrizable spaces and pairs. One of the key implications of our findings is that the -surjection property holds when the space is a union of homology cycles.

Our findings have further practical implications, one of which is that a finite simplicial complex X has computable type if and only if it satisfies the -surjection property for some > 0. Recall that this property says that every continuous function f : X → X which is -close to the identity must be surjective.

We establish that this property holds true if and only if every star, which takes the form of a cone C(L), satisfies the surjection property, namely every continuous function f : C(L) → C(L) which is the identity on L is surjective.

In addition, we prove that the computable type property does not remain invariant under products, providing an answer to a query posed by Čelar and Iljazović in [START_REF] Čelar | Computability of products of chainable continua[END_REF]. Specifically, we have discovered the existence of a finite simplicial complex X, which has computable type, but its product with the circle, X × S 1 does not. This intriguing result hinges on the intricate characteristics of the suspension homomorphism between homotopy groups of spheres.

By utilizing the reduction to homology, we can deduce that the computable type property can be effectively determined for finite simplicial complexes with dimension up to 4.

Furthermore, it implies that in a simplicial complex K, if every n-simplex belongs to an even number of maximal (n + 1)-simplices, then K has computable type.

The chapter is organized as follows. In Section 6.2, we introduce the notion of almost Euclidean spaces and pairs. In Section 6.3, we investigate general aspects of the ( -)surjection property, such as their preservation under countable unions. In Section 6.4, we show that for spaces that are finite unions of cones, the -surjection property is equivalent to the surjection property of each cone. In Section 6.5, we investigate the surjection property of cones and obtain precise relationships with homotopy and homology theories. In Section 6.6, we apply these results to a family of spaces which is a source of counter-examples. In Section 6.7, we give applications to the computable type property, the main of which is a characterization for finite simplicial complexes. We end with a conclusion in Section 6.8. Section B in the appendix discusses the choice of the coefficients in homology groups.

Euclidean Points and Regular Cells

We introduce a class of spaces for which most points have a Euclidean neighborhood. Most of the results will apply to these spaces. intersection A × B are ANRs, therefore their union is an ANR (Theorem IV.6.1 in [START_REF] Borsuk | Theory of retracts[END_REF]). We can assume w.l.o.g. that the metric on the product space is the maximum of the metrics on X and Y .

We assume that (X, A) does not satisfy the -surjection property for any > 0, and show that the same holds for (Z, C). Let > 0 and f : X → X be a non-surjective function such that f | A = id A and d(f, id X ) < . We naturally define g : (Z, C) → (Z, C) by g(x, y) = (f (x), y). One easily checks that g(C) ⊆ C. As d(f, id X ) < , one has d(g, id Z ) < . The image of g does not contain Z \ C: if x 0 ∈ X \ im(f ) and y 0 ∈ Y \ B, then (x 0 , y 0 ) / ∈ im(g). Therefore, we can apply Lemma 6.3.1, implying that (Z, C) does not satisfy the -surjection property for any > 0.

The ( -)Surjection Property for Cone Pairs

Cones have the particular property that they contain arbitrarily small copies of themselves, implying that for any > 0, the -surjection is equivalent to the surjection property. Cones and cone pairs are defined in Section 3.2.1. Proposition 6.3.3 Let (X, A) be a compact pair and > 0. The cone pair C(X, A) has the -surjection property iff it has the surjection property.

Proof. Assume that X is embedded in Q and let C(X) = {(t, tx) : t ∈ [0, 1], x ∈ X}. Assume that there is a non-surjective continuous function f : C(X) → C(X) which is the identity on X ∪ C(A). For any > 0, one can define such a function g which is -close to the identity. We decompose C(X) as C(X) = C ∪ D where

C = {(t, tx) : t ∈ [0, δ], x ∈ X} D = {(t, tx) : t ∈ [δ, 1], x ∈ X}.
Note that C is homeomorphic to C(X). We define g as the identity on D and as a rescaled version of f on C, namely g(δt, δtx) = δf (t, tx) for t ∈ [0, 1] and x ∈ X. g is non-surjective, is the identity on X ∪ C(A), and if δ is sufficiently small, then g is -close to the identity.

In Section 3.2.1, we introduce the symbol and define C( ) = {0}. Note that the proof works when A = and C(X, ) = (C(X), X ∪ {0}) (where 0 is the tip of C(X)).

The only case when we need to consider A = is when X is a singleton. Proposition 6.3.4

If X is not a singleton, then C(X, ) has the surjection property if and only if C(X, ∅) has the surjection property.

Proof. Assume that C(X, ∅) does not have the surjection property, and let f : C(X) → C(X) be a non-surjective continuous function which is the identity on X. We build a non-surjective continuous function g : C(X) → C(X) which is the identity on X ∪ {0}.

We are going to define a proper subspace Y C(X) that contains im(f ) ∪ {0} and contains a path from 0 to f (0). Let then i : X ∪ {0} → Y be the inclusion and j : X ∪ {0} → Y be the identity on X and send 0 to f (0). The path from 0 to f (0) in Y induces a homotopy from i to j. As the pair (C(X), X ∪ {0}) has the homotopy extension property, it implies that the inclusion has a continuous extension g : C(X) → Y , which is non-surjective when typed as g : C(X) → C(X), which completes the proof.

We now define the space Y . There are two cases. First assume that 0 ∈ im(f ). In that case, we simply take Y = im(f ). Let x ∈ C(X) be such that f (x) = 0. The image by f of the ray from 0 to x in C(X) is a path from f (0) to 0, contained in Y .

Finitely Conical Spaces

We show that for a class of spaces containing finite simplicial complexes and compact manifolds, the -surjection property is actually a local property.

Definition and First Properties

Definition 6.4.1. A topological space X is finitely conical if there exists a finite sequence of compact metrizable spaces (L i ) i≤n and a finite covering of X by open sets U i ⊆ X, i ≤ n, where U i is homeomorphic to OC(L i ). A pair (X, A) is finitely conical if X is finitely conical and for every i, the homeomorphism

f i : U i → OC(L i ) satisfies f (U i ∩ A) = OC(N i ) for some compact N i ⊆ L i (or N i = ).
We will say that the sequence (L i ) i≤n or (L i , N i ) i≤n witnesses the fact that X or (X, A) is finitely conical.

For simplicity of notation, we will identify U i with OC(L i ), so the condition for pairs can be written as OC(L i ) ∩ A = OC(N i ). Remark 6.4.1

We allow the pair (L i , N i ) = ({0}, ), giving the open cone pair OC({0}, ) = ([0, 1), {0}). Every compact manifold of dimension n ∈ N * with possibly empty boundary ∂M is finitely conical. The pairs (L i , N i ) are (S n-1 , ∅), as well as (B n-1 , S n-2 ) when ∂M = ∅. A point x ∈ M \ ∂M is the tip of OC(S n-1 ), a point x ∈ ∂M is the tip of OC(B n-1 ). Note that for n = 1, one has (B 0 , S -1 ) = ({0}, ) (see Section 3.2.1).

The class of finitely conical spaces or pairs is preserved by many constructs. Proposition 6.4.1

Finitely conical pairs are closed under finite products and the cone operator.

Proof. Let (X 1 , A 1 ) and (X 2 , A 2 ) be finitely conical, and let (L 1 i , N 1 i ) i≤m and (L 2 j , N 2 j ) j≤n be respective witnesses. It is not difficult to see that the (X, A) = (X 1 , A 1 ) × (X 2 , A 2 ) is finitely conical, witnessed by the pairs

(L i,j , N i,j ) = (L 1 i , N 1 i ) * (L 2 j , N 2 j )
, where the join * of pairs is defined in Section 3.2.2.

Indeed, one has

X = X 1 × X 2 = i OC(L 1 i ) × j OC(L 2 j ) = i,j OC(L 1 i ) × OC(L 2 j ) = i,j
OC(L i,j ), 2. ⇒ 1. We assume that for every > 0, (X, A) does not have the -surjection property, and we prove that some cone pair C(L i , N i ) does not have the surjection property.

As X is compact and is covered by finitely many open cones OC(L i ), X is covered by slightly smaller closed cones C(L i ) contained in these open cones. Let (K i , M i ) = C(L i , N i ) be these closed cones, contained in X. L i can be seen as a subset of the Hilbert cube Q, and there is a homeomorphism

g i : {(t, tx) : t ∈ [0, 1], x ∈ L i } → K i . sending {(1, x) : x ∈ L i } ∪ {(t, tx) : t ∈ [0, 1], x ∈ N i } to M i (if N i = , then the latter set is {(1, x) : x ∈ L i } ∪ {(0, 0)}).
For any t ∈ (0, 1), define the smaller copy (K i (t), M i (t)) of (K i , M i ) as follows:

K i (t) = g i ({(s, sx) : s ∈ [0, t], x ∈ L i }), M i (t) = (K i (t) ∩ A) ∪ g i ({(t, tx) : x ∈ L i }).
Let t 0 < 1 be such that the sets K i (t 0 ) cover X, which exists as X is compact. Pick two other real numbers t 0 < t 1 < t 2 < 1. Let > 0 be such that for all i,

N (K i (t 0 ), ) ⊆ K i (t 1 ), (6.1)
N (K i (t 1 ), ) ⊆ K i (t 2 ). (6.2)
Let δ < be smaller than the values provided by Lemma 3.3.3 applied to all the compact ANRs K i (t 2 ) and . By assumption, (X, A) does not have the δ-surjection property, i.e. there exists a non-surjective continuous function h : X → X such that h| A = id A and d X (h, id X ) < δ < . As X = i K i (t 0 ) and h is not surjective, there exists i such that

K i (t 0 ) h(X). (6.3) Let (K, M ) = (K i (t 2 ), M i (t 2 )) ∼ = C(L i , N i ).
We define a non-surjective continuous function G : K → K such that G| M = id M , showing that C(L i , N i ) does not have the surjection property. First observe that h(K i (t 1 )) is contained in K = K i (t 2 ), because h is -close to the identity, so h(K i (t 1 )) ⊆ N (K i (t 1 ), ) ⊆ K by (6.2). We define g :

K i (t 1 ) ∪ M → K by g| K i (t 1 ) = h| K i (t 1 ) and g| M = id M .
The function g is well-defined and continuous because h coincides with the identity on the intersection K i (t 1 ) ∩ M ⊆ A.

We now define a continuous extension G : K → K of g. Note that g is δ-close to the inclusion f :

K i (t 1 ) ∪ M → K and f has a continuous extension F := id K : K → K, so using Lemma 3.3.3, g has a continuous extension G : K → K satisfying d K (G, id K ) < . As G extends g, G| M = id M . It remains to show that G is not surjective. Indeed, K i (t 0 ) is not contained in G(K): G(K i (t 1 )) = h(K i (t 1 )) which does not contain K i (t 0 ) by (6.3), G(K \ K i (t 1 )) ⊆ N (K \ K i (t 1
), ) which is disjoint from K i (t 0 ) by (6.1).

The Surjection Property for Cone Pairs

Theorem 6.4.1 reduces the -surjection property to the surjection property for cone pairs. The purpose of this section is to develop techniques to establish whether a cone pair has the surjection property. We will only work with cones of almost Euclidean spaces.

Quotients to Spheres

The first characterization of the surjection property for cone pairs involves quotient maps to spheres associated to regular cells.

Let (X, A) be a compact pair and C ⊆ X \ A a regular n-cell. The quotient of X by X \ op(C) is homeomorphic to S n . We let q C : (X, A) → (S n , s) (6.4) be the quotient map, where s ∈ S n is the image of X \ op(C) by q C . Theorem 6.5.1. Let (X, A) be an almost Euclidean compact pair. The following statements are equivalent:

1. C(X, A) has the surjection property, 2. For every n ≥ 1 and every regular n-cell C ⊆ X \ A, the quotient map q C : (X, A) → (S n , s) is not null-homotopic.

Actually, the implication 1. ⇒ 2. holds without assuming that (X, A) is almost Euclidean. In order to prove the theorem, we first give a reformulation of the surjection property for cone pairs. Lemma 6.5.1

Let (X, A) be an almost Euclidean compact pair. The following statements are equivalent:

1. C(X, A) does not have the surjection property, 2. There exists a regular cell C ⊆ X \ A and a continuous function f : C(X) → X ∪ C(D),

where D = X \ op(C), which is the identity on X ∪ C(A).

Proof. 1. ⇒ 2. Let f : C(X) → C(X) be a non-surjective continuous function which is the identity on X ∪ C(A). The union of regular cells C ⊆ X \ A is dense in X \ A, so the union of their cones is dense in C(X \ A). Therefore, there exists a regular cell C ⊆ X \ A such that C(C) is not contained in the image of f . C(C) is a regular cell in C(X), its intersection with X ∪ C(A) is contained in its border. Let x belong to the interior of C(C) but not to the image of f . There exists a retraction r 0 : C(C)\ {x} → bd(C(C)), which extends to a retraction r : C

(X) \ {x} → X ∪ C(D). The composition r • f : C(X) → X ∪ C(D) is well-defined, continuous and is the identity on X ∪ C(A). 2. ⇒ 1. Such a function f is a non-surjective continuous satisfying f | C(A)∪X = id C(A)∪X .
The existence of a continuous function f : C(X) → X ∪ C(D) which is the identity on X is nothing else than a null-homotopy of the inclusion map i : X → X ∪ C(D). If the pair (X, D) has the homotopy extension property, then X ∪ C(D) is homotopy equivalent to X/D (Proposition 0.17 in [START_REF] Hatcher | Algebraic Topology[END_REF]), which implies that the inclusion map i is null-homotopic if and only if the quotient map q : X → X/D is null-homotopic. We prove a similar equivalence applicable to pairs (X, A). Essentially, we need to be careful about how the homotopy equivalence between X ∪ C(D) and X/D is defined on A.

0 1/2 1 0 1 t x 1 ∈ A x 2 / ∈ A x 3 / ∈ A Figure 6.1: Graph of t → h 1 (t, x), where h(t, x) = [h 1 (t, x), h 2 (t, x)], for different x's We claim that the function f : C(X) → X ∪ C(D) satisfying g = f • h is well-defined. Indeed, if h(u) = h(v), then u = v or u, v ∈ S, therefore g(u) = g(v).
As h is a quotient map and g is continuous, f is continuous as well.

The restriction of f to X (identified with {[0, x] : x ∈ X}) is the identity, because h| X = g| X = id X . The function h sends C(A) onto C(A), and coincides with g on C(A), so the restriction of f to C(A) is the identity.

Proof of Theorem 6.5.1. We combine Lemma 6.5.1 and Proposition 6.5.1, which is possible because the pair (X, X \ op(C)) has the homotopy extension property.

An interesting consequence is that it is always possible to replace a pair (X, A) by the single space X/A. Proof. Note that X/A is almost Euclidean, so we can apply Theorem 6.5.1 to both (X, A) and (Y, B) = (X/A, ∅). For each regular cell C ⊆ X \ A, the corresponding quotient map q : (X, A) → (S n , s) is null-homotopic if and only if the quotient map q : X/A → S n is null-homotopic, applying Proposition 3.3.1. Therefore, Theorem 6.5.1 applied to (X, A) and (Y, B) gives the equivalence.

Another consequence is that in Proposition 6.5.1, the condition that f : C(X) → C(X) is the identity on X ∪ C(A) can be equivalently replaced by an apparently weaker condition. is null-homotopic. It implies that the quotient map q : (X, A) → (X/D, p) is null-homotopic, which in turn implies the first condition by Proposition 6.5.1.

Retractions to Spheres

We give a simple sufficient condition implying that C(X, A) does not have the surjection property. We will later show that this condition is also necessary under certain assumptions. Theorem 6.5.2. Let (X, A) be a compact pair and C ⊆ X \ A be a regular n-cell. If there exists a retraction r : X \ op(C) → bd(C) which is constant on A, then the quotient map q : (X, A) → (S n , s) is null-homotopic, hence C(X, A) does not have the surjection property.

Note that a retraction r : X \ op(C) → bd(C) is the same thing as a retraction r : X → C which sends X \ op(C) to bd(C) (indeed, r can be obtained from r by extending it as the identity on C, and r can be obtained from r by restriction to X \ op(C)). Proof. Let D = X \ op(C) and r 0 : D → bd(C) be a retraction with constant value p on A. Let r : X → C be the retraction extending r 0 , defined as the identity on C. As a function of pairs, r : (X, A) → (C, p) is null-homotopic. Indeed, there is a contraction of C ∼ = B n that fixes p, i.e. the identity id C : (C, p) → (C, p) is null-homotopic, so r = id C • r is null-homotopic.

The quotient of (X, A) by D is (S n , s), and the quotient of (C, p) by bd(C) is (S n , s). Let q C : (X, A) → (S n , s) and q C : (C, p) → (S n , s) be the quotient maps. As r is the identity on C, one has q C = q C • r. Therefore, the null-homotopy of r induces a null-homotopy of q C . Theorem 6.5.1 implies that C(X, A) does not have the surjection property. Example 6.5.1

Let X be a topological graph and C be an edge. If C is neither contained in a cycle nor in a path from a point of A to another point of A (see an example on Figure 6.2), then C(X, A) does not have the surjection property because there is a retraction of X \ op(C) to bd(C), as we now explain. We will see with Corollary 6.5.4 that these conditions are tight.

As C does not belong to a cycle, its two endpoints a, b belong to two distinct connected components of X \ op(C). As there is no path from A to A through C, the connected component of a or b, say a, is disjoint from A. Let r : X \ op(C) → bd(C) send that component to a and all the rest to the other endpoint b. It is a retraction which is constant on A. The next result shows that the assumptions in Theorem 6.5.2 can be weakened. Proposition 6.5.2

Let (X, A) be a compact pair and C ⊆ X be a regular cell which is not contained in A. If there exists a retraction r : X \ op(C) → bd(C) whose restriction r| A : A → bd(C) is null-homotopic, then C(X, A) does not have the surjection property.

Proof. We show that there is a regular cell C ⊆ C \ A and a retraction r : X \ op(C ) → bd(C ) which is constant on A, and then apply Theorem 6.5.2. As C A and A is closed, there exists a regular cell C ⊆ op(C) \ A. The retraction r : X \ op(C) → bd(C) can be extended as the identity on C \ op(C ), giving a retraction r 1 : X \ op(C ) → C \ op(C ). There is a retraction r 2 : C \ op(C ) → bd(C ), because C \ op(C ) is a thickened sphere and bd(C ) is its inner sphere.

Let

f = r 2 • r 1 : X \ op(C ) → bd(C ). The restriction of f to A ∪ bd(C ) is homotopic to a function h : A ∪ bd(C ) → bd(C )
which is constant on A and is the identity on bd(C ) (extend the null-homotopy of r| A as the identity on bd(C ), which is disjoint from A). As bd(C ) is an ANR (it is indeed a sphere), Borsuk's homotopy extension theorem implies that f is homotopic to an extension of h. Such an extension is a retraction r : X \op(C ) → bd(C ) which is constant on A.

Cycles

Let (X, A) be a pair consisting of a topological finite graph X and a subset A of vertices. As a particular case of Corollary 6.5.4 proved below, C(X, A) has the surjection property if and only if every edge belongs to a cycle or a path between two points of A. It is assumed that cycles and paths visit each edge at most once. The result should be compared with Example 6.5.1.

More generally, we prove a higher-dimensional version for spaces like finite simplicial complexes, and even almost Euclidean spaces. The higher-dimensional notions of cycles and paths between points of A are the homology cycles and cycles relative to A, i.e. the elements of H n (X, A). We need to express the idea that a cell "belongs" to a relative cycle.

Recall that T = R/Z is the circle group. Definition 6.5.1. Let (X, A) be a compact pair and n ≥ 1. A regular n-cell C ⊆ X \ A belongs to a relative cycle if the canonical homomorphism

H n (X, A; T) → H n (X, X \ op(C); T) is non-trivial.
Let us explain why this definition is meaningful. First, we use the coefficient group T because it subsumes the more usual coefficients groups Z and Z/kZ (see Proposition B.0.1). The canonical homomorphism is non-trivial if there exists a relative cycle in (X, A) which is not cancelled when quotienting by the part of the space which is not in C, so the cycle should "visit" C. An alternative way of seeing this is to consider the exact sequence (with coefficients in T)

H n (X \ op(C), A) → H n (X, A) → H n (X, X \ op(C)).
The second homomorphism is non-trivial if and only if the first one is non-surjective, which means that there is a relative cycle in (X, A) which is not contained in X \ op(C). Remark 6.5.1

Here is an equivalent formulation of Definition 6.5.1, at least for pairs (X, A) where X and A are ANRs. C belongs to a relative cycle if and only if the canonical homomorphism on cohomology groups with coefficients in Z H n (X, X \ op(C)) → H n (X, A) is non-trivial. Indeed, H n (X, A; T) is the Pontryagin dual of H n (X, A) and the homomorphisms between homology and cohomology groups are dual to each other (Proposition G, §VIII.4, p. 137 and Proposition F, §VIII.5, p. 141 in [START_REF] Hurewicz | Dimension Theory (PMS-4)[END_REF]).

1) Simplicial Complexes

Let us illustrate Definition 6.5.1 in the case of finite simplicial complexes.

A finite simplicial complex is almost Euclidean, each maximal simplex being a regular cell. For simplicial complexes, singular homology is isomorphic to simplicial homology. Whether a simplex belongs to a relative cycle in the sense of Definition 6.5.1 can be reformulated in a much more explicit way using the concept of n-chains (for a comprehensive understanding of chains, you may refer to Section 2.1 in [START_REF] Hatcher | Algebraic Topology[END_REF]). Proposition 6.5.3

Let (X, A) be a finite simplicial pair and C a maximal simplex, of dimension n. C belongs to a relative cycle if and only if there exists an n-chain with coefficients in T (equivalently, either in Z or Z/kZ for some k ≥ 2) whose boundary is contained in A, assigning a non-zero coefficient to C. Proof. Let (∆ i ) i≤m be the n-simplices in X, and assume w.l.o.g. that ∆ 0 is C. Let D be the subcomplex obtained by removing C from X (but keeping its faces). The homomorphism

H n (X, A; T) → H n (X, D; T)
is realized, at the level of n-chains, by sending α i ∆ i to α 0 ∆ 0 . As ∆ 0 is maximal, ∆ 0 is not a boundary. Therefore, the homomorphism is non-trivial if and only if there exists a relative cycle assigning a non-zero coefficient α 0 to ∆ 0 .

2) Cycles vs the Surjection Property

Let (X, A) be a compact pair and C ⊆ X \ A be a regular n-cell (n ≥ 1). One has natural isomorphisms

H n (X, X \ op(C); T) ∼ = H n (C, bd(C); T) (by excision) ∼ = H n (C/bd(C), s; T) ∼ = H n (S n , s; T) ∼ = T.
Therefore, up to isomorphism, the canonical homomorphism from Definition 6.5.1

H n (X, A; T) → H n (X, X \ op(C); T)
is nothing else than the homomorphism induced by the quotient map q C : (X, A) → (S n , s) on homology groups. It has the following immediate consequences:

If C belongs to a relative cycle, then q C : (X, A) → (S n , s) is not null-homotopic, When dim(X) = n and X, A are ANRs, C belongs to a relative cycle if and only if q C : (X, A) → (S n , s) is not null-homotopic, by Hopf's classification theorem (Theorem 3.4.3). Therefore, we obtain partial generalizations of the claimed result for graphs in the beginning of the section to higher-dimensional spaces. The first one is only an implication. This result becomes an equivalence when the space "has the same dimension everywhere", giving a first extension from graphs to pure simplicial complexes, i.e. simplicial complexes X whose maximal simplices all have the same dimension. Corollary 6.5.4. (Cycles vs surjection property, equivalence) Let n ≥ 1. Let (X, A) be an almost n-Euclidean compact pair, where X and A are ANRs and dim(X) = n. The following statements are equivalent:

The pair C(X, A) has the surjection property, Every regular cell C ⊆ X \ A belongs to a relative cycle. Theorem 6.5.2 also becomes an equivalence when the dimension of the space matches the dimension of the cell, or the cell has dimension at most 2. Corollary 6.5.5. (Cycles vs retraction) Let (X, A) be a compact pair, where X and A are ANRs and dim(X) = n ≥ 1. For a regular n-cell C ⊆ X \ A, the following statements are equivalent:

C does not belong to a relative cycle,

There exists a retraction r : X \ op(C) → bd(C) which is constant on A.

The equivalence holds if n = 1 or 2, without any dimension assumption about X.

Proof. We assume that C does not belong to a relative cycle and build a retraction (the other implication is Theorem 6.5.2). Let Y = X/A. The existence of a retraction which is constant on A is equivalent to the existence of a retraction r : Y → bd(C). Let f : bd(C) → bd(C) be the identity and i : bd(C) → Y \ op(C) be the inclusion map. Note that bd(C) ∼ = S n-1 and let f * , i * be the induced homomorphisms on reduced homology groups H n-1 (•; T). We want to find a retraction, i.e. an extension of f . As dim(X) = n, we can apply Hopf's extension theorem (Theorem 3.4.2): f has an extension if and only if ker i * ⊆ ker f * . As f * is injective, f has an extension if and only if i * is injective.

In the exact sequence (with the coefficient group T)

H n (Y ) → H n (Y, Y \ op(C)) → H n-1 (Y \ op(C)) (6.5)
the first homomorphism is trivial by assumption (note that H n (X/A) ∼ = H n (X, A) as n ≥ 1), so the second homomorphism is injective. The latter is i * up to isomorphism, because one has natural isomorphisms

H n (Y, Y \ op(C)) ∼ = H n (C, bd(C)) ∼ = H n-1 (bd(C)).
As a result, i * is injective. It implies that f has an extension, which is the sought retraction.

The particular case n = 1 or 2 corresponds to the particular case in Hopf's extension theorem.

It also implies that one can extend from graphs to simplicial complexes of dimension at most 3 that are not necessarily pure.

Corollary 6.5.6. Let (X, A) be an almost Euclidean compact pair, where X and A are ANRs and dim(X) ≤ 3. The following statements are equivalent:

The pair C(X, A) has the surjection property, Every regular cell C ⊆ X \ A belongs to a relative cycle.

Proof. For every regular n-cell C, one has n = dim(X) or n = 1 or n = 2 so we can apply Corollary 6.5.5. If C does not belong to a relative cycle, then there exists a retraction r : X \ op(C) → bd(C), so the quotient map q C : (X, A) → (S n , s) is null-homotopic (Theorem 6.5.2).

We will see in Section 6.6 that the dimension assumption in Corollaries 6.5.4, 6.5.5 and 6.5.6 cannot be dropped.

We also obtain a relationship between cycles and the -surjection property for ANRs.

Theorem 6.5.3. Let (X, A) be an almost Euclidean compact pair, where X is an ANR. If every regular cell C ⊆ X \ A belongs to a relative cycle, then (X, A) has the -surjection property for some > 0.

Proof. As X is a compact ANR, if is sufficiently small, then every function f : X → X satisfying f | A = id A and d(f, id X ) < is homotopic to id X , via a homotopy h t : X → X such that h t | A = id A . We claim that (X, A) has the -surjection property. Assume otherwise, let f : X → X be non-surjective continuous, satisfy f | A = id A and d(f, id X ) < . f is homotopic to id X : (X, A) → (X, A).

Let C ⊆ X \ A be a regular n-cell which is disjoint from im f , i : (X \ op(C), A) → (X, A) the inclusion map and let f : (X, A) → (X \ op(C), A) be such that f = i • f . As f = i • f is homotopic to id X , i * • f * is an isomorphism on homology groups of (X, A), so i * : H n (X \ op(C), A) → H n (X, A) is surjective (coefficients are implicitly in T). In the exact sequence

H n (X \ op(C), A) i * -→ H n (X, A) -→ H n (X, X \ op(C))
the second homomorphism is therefore trivial, contradicting the assumption that C belongs to a relative cycle. Therefore, there is no such function f , so (X, A) has the -surjection property.

The converse implication in Theorem 6.5.3 does not hold, even under dimension assumptions, as the next example shows. Example 6.5.3

Let X be Bing's house (see Figure 6.10), or the house with two rooms, which is a pure 2dimensional simplicial complex, and A = ∅. We prove in Proposition 6.7.2 that it has thesurjection property, by using the fact that the star of each vertex in X is the cone of a graph which consists of a union of cycles. However, X is contractible, so no 2-cell belongs to a cycle. with itself using the homotopy group operation) is not null-homotopic, however its suspension S2h is (again by Corollary 4J.4 in [START_REF] Hatcher | Algebraic Topology[END_REF]). Therefore, in X 2h the quotient is null-homotopic and (C(X 2h ), X 2h ) does not satisfy the surjection property, but there is no retraction, so the implication 2. ⇒ 1. fails for X 2h .

B n+1 B m+1 S n+1 (a) The map q : X f → S n+1 p Sf B n+1 B m+1 S m+1 S n+1 (b) The map Sf • p : X f → S n+1 id C(f ) B n+1 B m+1 B n+1 B n+1 (c) The map k : X f → S n+1

Product

We have shown with Proposition 6.3.2 that for finite simplicial pairs, if (X, A)×(Y, B) satisfies thesurjection property for some > 0, then both (X, A) and (Y, B) satisfy the δ-surjection property for some δ > 0.

We prove that the converse implication does not hold.

Theorem 6.6.2. There exists a finite simplicial pair (X, A) that has the surjection property, but the product (X, A) × (B 1 , S 0 ) does not have the -surjection property for any > 0. There exists a finite simplicial complex Y that has the -surjection property for some > 0, but the product Y × S 1 does not have the δ-surjection property for any δ > 0.

Proof. We choose m, n ∈ N and f : S m → S n so that Sf is not-null-homotopic but S 2 f is (we will give detailed references in the next section below on the existence of such functions). We then topological spaces, including common compact manifolds and geometric models in computer graphics. Moreover, finite simplicial complexes can be effectively described using finite combinatorial information, enabling us to potentially achieve a comprehensive characterization of their computable type.

We focus solely on finite simplicial complexes, as infinite simplicial complexes, under the usual topologies, become non-compact.

We establish two topological characterizations for simplicial pairs (X, A) that have computable type. One characterization operates on a global level (the -surjection property), while the other offers a local perspective (the surjection property).

The local characterization simplifies the process of determining whether a simplicial pair (X, A) has computable type, as it involves inspecting the neighborhoods of each vertex individually (see Figure 6.6). Figure 6.6: A cone at a vertex in a finite simplicial complex Previous techniques found in the literature were too specialized to be applied to these sets. Our techniques not only accommodate any simplicial complex but also provide a straightforward and visual approach to addressing the computable type question for numerous sets.

By combining all the results, we arrive at the following comprehensive characterization for finite simplicial pairs. Corollary 6.7.1. For a finite simplicial pair (X, A) such that A has empty interior in X, the following statements are equivalent:

1. (X, A) has strong computable type, 2. (X, A) has computable type, 3. (X, A) satisfies the generalized -surjection property for some > 0, 4. (X, A) satisfies the -surjection property for some > 0, 5. For every vertex, the cone pair corresponding to its star has the surjection property (see Section 3.5).

Proof. The following implications hold for any compact pair: (1) ⇒ (2) and ( 3) ⇒ (4). Let us prove that (2) ⇒ (4) by contra-position. Note that the standard realization of (X, A) is computable. We show that if it does not have the -surjection property for any > 0, then it has computable witnesses, which together with Theorem 4.5.2 concludes the proof of (2) ⇒ (4). Claim 6.7.1

If a simplicial pair (X, A) does not have the -surjection property for any > 0, then its standard realization has computable witnesses.

Proof of Claim 6.7.1. By (2) ⇒ (1) in Theorem 6.4.1, there exists a cone pair (K i , M i ) = C(L i , N i ) which does not have the surjection property, so there exists a non-surjective function f 0 : K i → K i such that f 0 | M i = id M i . One can assume w.l.o.g. that d X (f 0 , id X ) < 1. Let δ 0 > 0 be such that d H (f 0 (X), X) > δ 0 . Given > 0, the number δ = δ 0 can be computed from and is anwitness. Indeed, the function f obtained by applying f 0 to a version of K i scaled by a factor and extended as the identity elsewhere satisfies all the conditions. Let us prove that (4) ⇒ (1).

We now prove the announced implication. Assume that (X, A) is embedded as a semicomputable pair in Q and has the -surjection property for some > 0. X can be subdivided so that each simplex has diameter less than /4 (for instance by barycentric subdivision, see [START_REF] Hatcher | Algebraic Topology[END_REF]). Let (M i ) 1≤i≤n be the maximal simplices of X and ∂M i the union of the proper faces of M i . Let Y i = (X \ M i ) ∪ ∂M i . One has A ⊆ Y i and Y i is an ANR because it is a finite simplicial complex.

Let α i > 0 be provided by Lemma 3.3.3 applied to Y i and 4 , and let α = min i (α i ). Using Lemma 5.2.1, for every i, let V i ⊇ Y i be a finite union of rational balls and r i : V i → Y i a retraction such that d V i (r i , id V i ) < min( /4, α/2). Let (f j ) j∈N be a dense computable sequence of functions from Q to itself provided by Lemma 4.3.2. Now, let U ⊆ Q be an open set and Z = (X \ U ) ∪ A. Claim 6.7.2

The following are equivalent:

1. U intersects X, 2. There exist i ≤ n and a continuous function g : Z → Y i such that g| A = id A and d Z (g, id Z ) < /4, 3. There exist i ≤ n and j such that f j (Z) ⊆ V i , d A (f j , id A ) < α 2 and d Z (f j , id Z ) < /2.

Proof of Claim 6.7.2. 1 ⇒ 2: Let i ≤ n be such that M i ∩ U = ∅. Take x ∈ U ∩ M i \ ∂M i and a retraction r : X \ {x} → Y i . Let g be the restriction of r to Z. g satisfies the conditions because A ⊆ Y i and the diameter of M i is less than /4.

2 ⇒ 3: Note that the conditions that f j should satisfy are satisfied by g and by any function that is sufficiently close to g. As the sequence (f j ) j∈N is dense, one can take f j arbitrarily close to g, so that f j satisfies the required conditions.

3 ⇒ 1: Suppose that U is disjoint from X, i.e. Z = X. Let F = r i • f j : X → Y i . One has d A (F, id A ) = d A (r i • f j , id A ) ≤ d A (r i • f j , f j ) + d A (f j , id A ) < α/2 + α/2 = α. Therefore using Lemma 3.3.3 there exists a continuous extension G : X → Y i of id A such that d X (G, F ) < /4. One has d X (G, id X ) ≤ d X (G, r i • f j ) + d X (r i • f j , f j ) + d X (f j , id X ) < /4 + /4 + /2 = , which contradicts the -surjection property of (X, A).

If U is a rational ball in Q, then 3. is semidecidable. As 3. is equivalent to 1., one can semidecide which rational balls U intersect X, therefore X is computable. Remark 6.7.1

For a finite simplicial pair that is itself homeomorphic to a cone pair i.e. (X, A) = C(L, N ), we obtain a further equivalence, since the surjection and -surjection properties are equivalent for cone pairs. Remark 6.7.2

The equivalence between the -surjection property and strong computable type can be done more generally for compact ANRs that are covered by finitely many regular cells. However, the proof does not use the existence of computable retractions, since such spaces need not to have a computable copy; one needs to generalize our proof of implication 2. ⇒ 1. of Theorem 3.4 in [START_REF] Djamel | Computability of finite simplicial complexes[END_REF] to these spaces to obtain the result. Remark 6.7.3

A single topological space X has many different simplicial decompositions, i.e. many abstract simplicial complexes whose realizations are homeomorphic to X. For instance, a triangle can be decomposed into many smaller triangles. At first sight, the fifth condition in Corollary 6.7.1 depends on the choice of the decomposition, because the local cone pairs are taken at the vertices of the decomposition. However, the theorem implies that the choice of the simplicial decomposition is irrelevant, because the other conditions do not depend on the decomposition: if all the cone pairs in a simplicial decomposition have the surjection property, then it is still true for all other simplicial decompositions of the space.

The next Corollary is a direct consequence of Corollary 6.7.1 and Theorem 6.3.2. Corollary 6.7.2. Let (X, A) be a finite simplicial pair and (X i , A i ) i≤n be pairs of subcomplexes such that X = i≤n X i and A = i≤n A i . If every pair (X i , A i ) has computable type, then (X, A) has computable type.

For instance, if a finite simplicial complex X is a finite union of subcomplexes that are homeomorphic to spheres, then X has computable type. More generally, if a finite simplicial pair (X, A) is a finite union of pairs of subcomplexes (X i , A i ) that are homeomorphic to pairs ( § n , ∅) or (B n+1 , § n ), then (X, A) has computable type. Corollary 6.7.3. For finite simplicial pairs (X, A) such that X is pure or has dimension at most 4, whether (X, A) has computable type is decidable.

Proof. Such a pair (X, A) is finitely conical and has computable type if and only if each cone pair C(L i , N i ) has the surjection property. If X is pure or has dimension at most 4, the each L i is pure or has dimension at most 3, so the surjection property for C(L i , N i ) is equivalent to the fact that each maximal simplex of L i belongs to a relative cycle in (L i , N i ) by Corollaries 6.5.4 and 6.5.6. As homology groups of finite simplicial complexes can be computed, this property is therefore decidable.

Finite 2-Dimensional Simplicial Complexes

We present a comprehensive characterization of 2-dimensional simplicial pairs with computable type (note that they are locally cones of finite graphs). This characterization is achieved by leveraging the surjection property of local cone pairs and reducing it to a simple graph property (Corollary 6.7.4), which comes from Corollaries 6.5.4 and Remark 6.7.1.. Consequently, we provide a visual tool that effectively demonstrates the computable type of finite 2-dimensional simplicial complexes. To further emphasize the strength of our findings, we showcase non-trivial applications through the examination of two famous sets: the dunce hat (Figure 6.9a) and Bing's house (Figure 6.10). Corollary 6.7.4. Let (L, N ) be a pair such that L is a finite graph and N is a subset of its vertices. The following statements are equivalent:

1. C(L, N ) has the surjection property, 2. Every edge is in a cycle or a path starting and ending in N , 3. C(L, N ) has computable type.

We follow the usual convention that in a graph, a path and a cycle do not visit a vertex twice, i.e. they are topologically a line segment and a circle respectively. In particular, a path connects two different points. Example 6.7.1

Fix some n ≥ 1 and let X be the star with n branches and A be the n endpoints of these branches (see Figure 6.7), with a special case for n = 1: C({v}, ) = (B 1 , § 0 ). The pair (X, A) is precisely C(A, ∅). As A has no edge, it satisfies the conditions of Theorem 6.7.4, therefore (X, A) has the surjection property. One can then obtain Iljazović's result that every finite graph has computable type [START_REF] Iljazović | Computability of graphs[END_REF], because the local cones of a finite graph are stars, which have the surjection property. Fix some n ≥ 2 and let X be the union of n squares which all meet in one common edge and A be the union of all the other edges (see Figure 6.8). The pair (X, A) has the surjection property. Indeed, (X, A) = C(A, ∅) and A is a graph which is a union of circles (each circle is the boundary of the union of two squares). Therefore, C(A, ∅) has the surjection property and computable type by Corollary 6.7.4 1) The Dunce Hat Definition 6.7.1. The dunce hat D is the space obtained from a solid triangle by gluing its three sides together, with the orientation of one side reversed (see Figure 6.9a). complex structure (by triangulating each flat surface). It is then a 2-dimensional simplicial complex with no free edge, which means that every edge belongs to at least two triangles. Using our results we easily show that this set has computable type as a single set, i.e. without adjoining a boundary to it. Proposition 6.7.2

Bing's house has computable type.

It is worth noticing that thanks to our results, it can be proved by looking at pictures only, although the argument can be formalized. Proof. Using Corollary 6.7.1, it is sufficient to inspect the possible local cones. One easily sees that there are three types of possible cones, depicted in Figure 6.11. The basis of each cone is a graph which is a union of 1, 2 or 3 cycles, so by Corollary 6.7.4 each cone pair has the surjection property, therefore Bing's house has computable type by Corollary 6.7.1.

Boundaries and the Odd Subcomplex

Given a simplicial complex X, a natural problem is to understand whether there is a minimal notion of boundary ∂X such that the pair (X, ∂X) has computable type. We make a few observations about three possible candidates. Let ∂ 1 X be the union of simplices that are contained in exactly one simplex of the next dimension, i.e. ∂ 1 X is the union of the free simplices of X, ∂ + X be the union of simplices that are contained in at least one simplex of the next dimension, ∂ odd X be the union of simplices that are contained in an odd number of maximal simplices of the next dimension. Proposition 6.7.3

Every simplicial pair (X, ∂ + X) has computable type.

Proof. Let (M i ) i≤n be an enumeration of the maximal simplices of X. M i is a ball, let ∂M i be its bounding sphere, which is a subcomplex of M i . One has X = i≤n M i and ∂ + X = i≤n ∂M i . Each pair (M i , ∂M i ) has the surjection property (Example 6.3.1), so (X, ∂ + X) has the -surjection property for some by Theorem 6.3.2. As a result, (X, ∂ + X) has computable type by Corollary 6.7.1. Proposition 6.7.4

Let X be a finite simplicial complex and A a subcomplex. If (X, A) has computable type, then A contains ∂ 1 X.

Proof. Assume that some simplex ∆ belongs to ∂ 1 X but not to A. We show that for every > 0, (X, A) does not have the -surjection property, implying that (X, A) does not have computable type by Corollary 6.7.1. Let > 0. Let ∆ be the unique maximal simplex having ∆ as a face (∆ has one more vertex than ∆). There is a non-surjective function f : ∆ → ∆ which is -close to the identity and is the identity on the other faces of ∆ : f slightly pushes points of ∆ away from ∆. We extend f as the identity on the rest of X, which gives a continuous function because ∆ is free. As ∆ is not in A, f is the identity on A.

The Odd Subcomplex

If K is a finite simplicial complex, then there is a natural subcomplex A of K such that the pair (K, A) has the -surjection property for some > 0. This subcomplex is the odd subcomplex of K. Definition 6.7.2. Let K be a finite simplicial complex of dimension m. For 0 ≤ n < m, the odd n-subcomplex ∂ n odd (K) of K is the collection of all n-simplices of K that are contained in an odd number of maximal (n + 1)-simplices in K. The odd subcomplex ∂ odd (K) of K is the union 0≤n<m ∂ n odd (K).

Clearly, for every finite simplicial complex K, ∂ odd (K) has empty interior because it contains no maximal simplex of K. Theorem 6.7.1. For every finite simplicial complex K, the pair (K, ∂ odd (K)) has thesurjection property for some > 0, hence has computable type.

Proof. Any finite simplicial complex is an ANR, so the pair (K, ∂ odd (K)) has the assumption of Theorem 6.5.3, therefore it is sufficient to show that every maximal simplex in K belongs to a relative cycle. We use the formulation of this latter property provided by Proposition 6.5.3.

Let S be a maximal n-simplex of K. Let c be the formal sum of the maximal n-simplices of K, seen as an n-chain with coefficients in Z/2Z. The boundary of c is the sum of the (n -1)-simplices that are contained in an odd number of maximal n-simplices. Therefore, ∂c is a chain of ∂ odd (K) so c is a relative cycle in (K, ∂ odd (K)). As S is itself a maximal n-simplex, its coefficient in c is 1, so S belongs to a relative cycle and the proof is complete.

The following observations can be made: Although (X, ∂ 1 X) has computable type when X is a 1-dimensional complex (i.e., a graph), it is no more true for 2-dimensional complexes. For the dunce hat D, one has ∂ 1 D = ∅ but we saw in Proposition 6.7.1 that (D, ∅) does not have computable type.

While (X, ∂ + X) always has computable type by Proposition 6.7.3, ∂ + X is far from optimal. For instance, it is always non-empty (unless X is a single point), but for any sphere S n , the pair (S n , ∅) already has computable type. (X, ∂ odd X) always has computable type. Observe that ∂ odd X is in general not optimal, as the example of graphs shows: (X, ∂ 1 X) has computable type and ∂ 1 X is usually smaller than ∂ odd X, which contains all the vertices of odd degrees.

Computable Type and the Product

Whether the computable type property is preserved by taking products was an open question, raised by Čelar and Iljazović in [START_REF] Čelar | Computability of products of chainable continua[END_REF]. Theorem 6.6.2 enables us to give a negative answer to this question. Note that (B 1 , S 0 ) and S 1 both have computable type. Corollary 6.7.5. There exists a finite simplicial pair (X, A) that has computable type, but such that the product (X, A) × (B 1 , S 0 ) does not. There exists a finite simplicial complex Y that has computable type, but such that the product Y × S 1 does not.

Proof. For simplicial spaces and pairs, computable type is equivalent to the -surjection property for some , so we can simply apply Theorem 6.6.2.

Note however that the converse direction holds: we proved that if (X, A) × (Y, B) has computable type, then both (X, A) and (Y, B) have computable type (assuming that they both have a semicomputable copy, which is the case for finite simplicial pairs), see Proposition 4.4.1. This implication is consistent with Proposition 6.3.2, which shows that the -surjection property behaves similarly.

Cones of Manifolds

Let (M, ∂M ) be a compact n-manifold with (possible empty) boundary. It is almost n-Euclidean, because every point of M \ ∂M is n-Euclidean.

One has H n (M, ∂M ; Z/2Z) ∼ = Z/2Z and the fundamental homology class intuitively contains every point of M \ ∂M . More formally, every regular n-cell C ⊆ M \ ∂M belongs to a relative cycle with coefficients in Z/2Z (therefore in T), in the sense of Definition 6.5.1. Indeed, M \ ∂M is Z/2Z-orientable so the homomorphism H n (M, ∂M ; Z/2Z) → H n (M, M \ op(C); Z/2Z) ∼ = Z/2Z is an isomorphism, sending the fundamental class to the generator of Z/2Z (see [START_REF] Hatcher | Algebraic Topology[END_REF] for instance).

Therefore, we can apply Corollary 6.5.3 and Remark 6.7.2, giving the next result.

Perspectives

Open questions could be considered as future directions, our study leaves many of them, we give here some examples.

Open Questions about Strong Computable Type

The example from Section 5.4.4 is a space that has strong computable type and properly contains copies of itself. The argument strongly relies on the fact that it contains the Hilbert cube, which is infinite-dimensional.

Question 6.8.1. Is there a finite-dimensional space which has strong computable type and contains a proper copy of itself? Question 6.8.2. Theorem 4.4.2 assumes that the pair is minimal for some Σ 0 2 invariant. Can this assumption be dropped? Is it always true that if (X, A) has strong computable type, then SCT (X,A) ≤ t W C N ? Question 6.8.3. Is there a compact space having (strong) computable type and infinitely many connected components?

We have revisited several results from the literature on computable type, identifying a Σ 0 2 invariant for which the space or the pair is minimal, we do not know whether the results on finite simplicial complexes (Corollary 6.7.1) can be restudied in a similar way. Question 6.8.4. If a finite simplicial pair (X, A) has strong computable type, is it minimal satisfying some Σ 0 2 topological invariant? Is there a canonical notion of boundary ∂X for a simplicial complex X, such that (X, ∂X) always has computable type, and ∂X is minimal in some sense?

Open Questions about the Descriptive Complexity of Topological Invariants

To any compact space X is associated the problem of recognizing X:

Input. A compact space Y ,
Output. Is Y homeomorphic to X? Note that each particular space X induces a topological invariant, which is "being homeomorphic to X". The obvious descriptive complexity of this invariant is Σ 1 1 , because it is formulated using an existential quantifier over continuously many objects: Y is homeomorphic to X iff there exists a homeomorphism f : Q → Q such that f (Y ) = X. However, a classical theorem from Descriptive Set Theory, due to Miller ([44], Theorem 15.14) and Ryll-Nardzewski ( [START_REF] Becker | The Descriptive Set Theory of Polish Group Actions[END_REF], Theorem 2.3.4) implies that the complexity of recognizing a particular space is always Borel. For the interested reader, this result states that when a Polish group continuously acts on a Polish space, each orbit is Borel. In our context, the Polish group of homeomorphisms from Q to itself continuously acts on the Polish space of compact subsets of Q, and the set of copies of a space is an orbit of the group action, and is therefore Borel by this theorem.

This result opens up the following research program: for each concrete space X, what is the exact complexity of recognizing X?

The results presented in this chapter are the first steps of a much broader research program. We list a few important problems left for future investigations.

We have built ad hoc Σ 0 2 invariants separating finite topological graphs.

Question 6.8.5. Is there a "natural" class of Σ 0 2 invariants separating finite topological graphs?

The family of finite graphs is very restricted. The finite simplicial complexes, which are generalizations of graphs to higher-dimensions, provide a rich family of spaces. Question 6.8.6. For each n ∈ N, what is the minimal level of complexity of topological invariants that can separate any pair of n-dimensional finite simplicial complexes? In the 1dimensional case of graphs, we saw that it is Σ 0 2 .

Question 6.8.7. What is the descriptive complexity of recognizing the circle? the disk? The same question can be raised for any particular compact space. In M , ∂M has a collar neighborhood, i.e. an open neighborhood that is homeomorphic to ∂M × [0, 1), in which ∂M is identified with ∂M × {0} (Proposition 3.42 in [START_REF] Hatcher | Algebraic Topology[END_REF]). Moreover we can assume that this collar neighborhood does not contain x (otherwise take the subset that is sent to ∂M ×[0, ) by the homeomorphism, for sufficiently small > 0). For simplicity of notation, we identify the collar neighborhood with ∂M × [0, 1). As the collar neighborhood deformation retracts to ∂M , one has for any abelian group G, H n (M x , ∂M ; G) ∼ = H n (M x , ∂M × [0, 1); G) ∼ = H n (M x \ ∂M, ∂M × (0, 1); G)

Index

where the last isomorphism exists by excision (it is a classical derivation that can be found for instance in [START_REF] Bredon | Topology and Geometry[END_REF], at the beginning of section VI.9). Similarly, H n (M, ∂M ; G) ∼ = H n (M \ ∂M, ∂M × (0, 1); G). We apply Theorem A.1.2 to the manifolds N = M \ ∂M and N x = M x \ ∂M and their subsets A = N \ (∂M × (0, 1)) and A x = N x \ (∂M × (0, 1)). A is compact but A x is not because x has been removed, so by Theorem A.1.2,

H n (M \ ∂M, ∂M × (0, 1); G) = H n (N, N \ A) 0.

H n (M x \ ∂M, ∂M × (0, 1); G) = H n (N x , N x \ A x ) ∼ = 0.

As a result, H n (M, ∂M ; G) 0 and H n (M x , ∂M ; G) ∼ = 0.

We can now prove that compact connected n-manifolds with boundary are H n -minimal, using the same argument as for manifolds without boundary. If x ∈ M \ ∂M and B ⊆ M \ ∂M is an open Euclidean ball around x, then M x deformation retracts to M \B, which means that there is a retraction r : M x → M \B such that if i : M \B → M x is the inclusion map, then r • i is homotopic to id Mx , and the homotopy is constant on M \ B. It is a classical result that deformation retractions preserve homology groups (Proposition 2.19 in [START_REF] Hatcher | Algebraic Topology[END_REF]), so H n (M \ B, ∂M ; G) ∼ = H n (M x , ∂M ; G) ∼ = 0.

Theorem 5.3.3 relates H n with homology groups, but is only for single sets. However, pairs can be reduced to single sets as follows. One the one hand, relative homology groups of pairs satisfying the homotopy extension property are isomorphic to homology groups of their quotients, so:

H n (M/∂M ; G) ∼ = H n (M, ∂M ; G) 0, H n (M x /∂M ; G) ∼ = H n (M x , ∂M ; G) ∼ = 0.

On the other hand, a pair (X, A) is in H n iff X/A is in H n (Proposition 5.3.10).

Therefore, Theorem 5.3.3 implies that (M, ∂M ) ∈ H n and (M \ B, ∂M ) / ∈ H n . It implies that (M, ∂M ) is H n -minimal by Lemma 5.3.2: if (X, A) is a proper compact subspair of M , then (X, A) ⊆ (M \ B, ∂M ) for some B ⊆ M \ ∂M , so (X, A) / ∈ H n .

  (a) La soucoupe de Varsovie (b) Une demi-coupe de la soucoupe de Varsovie

Figure 1 :

 1 Figure 1: La soucoupe de Varsovie et une demi-coupe de celui-ci

Figure 2 :

 2 Figure 2: La maison à deux pièces de Bing, ainsi qu'une demi-coupe

Figure 3 :

 3 Figure 3: Un disque attaché à un tore pincé n'est pas de type calculable

Figure 4 :

 4 Figure 4: Une copie semi-calculable d'un disque attaché à un tore pincé qui n'est pas calculable

  (a) The Warsaw saucer (b) A half-cut of the Warsaw saucer

Figure 5 :

 5 Figure 5: The Warsaw saucer and a half-cut of it

  xxiii

Figure 6 :

 6 Figure 6: Bing's house with two rooms and a half-cut of it

Figure 7 :

 7 Figure 7: A disk attached to a pinched torus does not have computable type

Figure

  Figure 8: A semicomputable copy of a disk attached to a pinched torus which is not computable

Definition 2 ., 2 . 3 .

 223 0.1. (Enumeration reducibility) Let A, B ⊆ N. We say that A is enumeration reducible to B, written A ≤ e B, if one of the following equivalent statements holds:1. There is an effective procedure producing an enumeration of A from any enumeration of BThere exists a c.e. set W ⊆ N such that, for all x ∈ N,x ∈ A ⇐⇒ there exists a finite set D ⊆ B such that x, D ∈ W, For every oracle O ⊆ N, if B is c.e. relative to O then A is c.e. relative to O.

Example 2 . 1 . 1 R

 211 equipped with the topology consisting of all rational open intervals is an effective countablybased T 0 -space.

Example 2 . 1 . 2

 212

Example 2 . 1 . 3

 213 Let n ∈ N * . R n equipped with the topology consisting of all rational open balls is a computable metric space.Remark 2.1.1

Fact 2 . 1 . 1

 211

Fact 2 . 2 . 1

 221

Contents

3. 1

 1 Point-Set Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 Cone and join . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2.1 Cone and Cone Pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Special Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2.2 Join . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Join of pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 Homotopy and Absolute Neighborhood Retracts (ANRs) . . . . . . . . . 33 3.3.1 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3.2 Homotopy extension property . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3.3 Absolute Neighborhood Retracts (ANRs) . . . . . . . . . . . . . . . . . . 34 3.3.4 ANRs and Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3.5 Pair vs quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4 Homology and Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4.1 Hopf's extension and classification theorems . . . . . . . . . . . . . . . . . 37 3.5 Simplicial Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Definition 3 . 1 . 3 .

 313 If (X, A) and (Y, B) are two pairs, then a function between pairs f :

Definition 3 . 1 . 5 .

 315 A manifold of dimension n, or more concisely an n-manifold, is a Hausdorff space M in which each point has an open neighborhood homeomorphic to R n . An n-manifold with boundary is a Hausdorff space M in which each point has an open neighborhood homeomorphic either to R n or to the half-space R n + = {(x 1 , . . . , x n ) ∈ R n : x n ≥ 0}. The set of points which have an open neighborhood homeomorphic to the half-space is called the boundary of M and is denoted by ∂M . Example 3.1.5 (B n+1 , S n ) is a compact manifold with its boundary, S n , the torus and Klien bottle are closed manifolds without boundary.

Theorem 3 .

 3 3.1. (Borsuk's homotopy extension theorem) Let Y be an ANR and (X, A) be a pair. If f, g : A → Y are continuous and f has a continuous extension F : X → Y , then every homotopy between f and g can be extended to a homotopy between F and some continuous extension G : X → Y of g. The next lemma is an application of Borsuk's homotopy extension theorem (see Exercise 4.1.5 in [60]). Lemma 3.3.3

  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2 Computable Type and the Hilbert Cube . . . . . . . . . . . . . . . . . . . . 44 4.3 Strong Computable Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.3.2 A First Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.3.3 A Characterization using Invariants . . . . . . . . . . . . . . . . . . . . . . 49 4.4 Strong Computable Type: Further Properties . . . . . . . . . . . . . . . . 52 4.4.1 Product and Cone of Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.4.2 A Topological Necessary Condition . . . . . . . . . . . . . . . . . . . . . . 53 4.4.3 The -Surjection Property . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.4.4 Uniformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.4.5 Discussion: Vietoris vs Upper Vietoris Topology . . . . . . . . . . . . . . 57 4.5 Failing to have Computable Type . . . . . . . . . . . . . . . . . . . . . . . . 58 4.5.1 Computable Witnesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Comparing Computability in Three Topologies . . . . . . . . . . . . . 58 Application to Computable Type . . . . . . . . . . . . . . . . . . . . 59 4.5.2 Simple copies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.5.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Definition 4 . 2 . 1 .

 421 An effective countably-based T 0 -space (X, τ ) is computably regular if there is a computable procedure associating to each basic open set B n a sequence of effective open sets (U k ) k∈N and a sequence of effective closed sets (

  Lemma 4.2.1

  We first show that U is effectively open in τ 2 upV . If σ = σ 0 . . . σ k is a finite sequence of basic τ 2 upVopen sets and U is a basic open subset of Q, then let

  a basis of the topology of X consisting of open sets, then the family {(X \ B, A \ B) : B ∈ B, B ∩ X = ∅} is cofinal, If A has empty interior in X, then the family {(Y, A) : A ⊆ Y X} is cofinal.

Remark 4 . 4 . 1

 441 The other implication of Proposition 4.4.2 is false. Let L be the graph consisting of two circles joined by a line segment, and N = ∅ (see Figure4.1). The pair (L, N ) has strong computable type but C(L, N ) does not, as shown in Proposition 6.7.1, because the graph L has an edge which belongs to no cycle.

  v

Figure 4 . 1 :Figure 4 . 2 :

 4142 Figure 4.1: Cone of two circles joined by a segment

Remark 4 . 5 . 1

 451

Figure 4 . 4 :

 44 Figure 4.4: A disk attached to a pinched torus and its -deformations

Figure 4 . 5 :

 45 Figure 4.5: A semicomputable copy of a disk attached to a pinched torus which is not computable

67 5. 2

 672 Computability-Theoretic Properties of ANRs and [G] X = [F ] X , then G is homotopic to an extension of g by Borsuk's homotopy extension theorem.

Claim 5 . 2 . 1

 521 The set {i ∈ dom(ν [X;Y ] ) : ν [A;Y ] (i) has a continuous extension to X} is c.e. relative to X and A.

Lemma 5 . 3 . 3

 533 Let (X, A) be a compact pair and Y an ANR. Every continuous function f : X ∪ Cyl(A) → Y has a continuous extension to Cyl(X). Every continuous function g : C(A) → Y has a continuous extension to C(X).

Figure 5 . 1 :

 51 Figure 5.1: Consecutive extensions of f

  ) is covered by the two disjoint open sets U a := i<k C i and U b := i>k C i , containing a and b respectively.

Figure 5 . 2 :

 52 Figure 5.2: The Warsaw saucer and a half-cut of it

4 .

 4 If X, Y are connected and X is non-trivial, then X approximates Y .Proof. (1) It follows from Proposition 5.5.2.

Figure 5 . 3 :

 53 Figure 5.3: -deformations of I converging to a copy of S

5 . 7

 57 It results from the following simple observations, left to the reader:In a graph G = (V, E), the sum S = v∈V :deg(v)≥3 deg(v) does not change when subdividing and does not increase when contracting an edge. Indeed, if the endpoints of the contracted edge have degrees a, b, then the new vertex has degree a + b -2, and a simple case analysis (a, b = 1, 2 or ≥ 3) shows that S cannot increase; in addition, the sum S is a topological invariant, i.e. does not change under subdivisions. Therefore, S must be the same in G, H and all the intermediate graphs of the contractions, If an edge contraction does not change S, then the resulting graph is homeomorphic to the original graph. If at least one of the endpoints of the edge has degree 2, then the graphs are homeomorphic. If one of the endpoints has degree 1 and the other has degree d ≥ 3, then 93 Separating Finite Topological Graphs they are replaced with a vertex of degree d -1, so S decreases by 1 or d. If both endpoints have degrees a, b ≥ 3, then they are replaced with a vertex of degree a + b -2, so S decreases by 2. If both vertices have degree 1, then one of the graphs is an edge, the other is a vertex, but a point cannot be contracted to any other graph. Remark 5.7.1

Figure 5 . 5 :

 55 Figure 5.5: An impossible case: x b and x b are close, but not closely connected

6. 1

 1 Introduction T he -surjection property is a topological property of spaces and pairs of spaces, introduced in Section 4.4.3 as a mean to establish a prerequisite for having strong computable type (see Definition 4.4.3 and Corollary 4.4.3). The surjection property is stronger since it does not depend on .

Example 6 . 4 . 1 Let

 641 X be a finite topological graph and A be the set vertices of degree 1. The pair (X, A) is finitely conical. An open cone consists of a vertex v together with the open edges starting at v. The open cone pair centered at v is OC(L, N ), where L is the set of vertices that are neighbors of v and N = if v has degree 1, and N = ∅ otherwise. Example 6.4.2 More generally, every pair (X, A) consisting of a finite simplicial complex X and a subcomplex A is finitely conical, witnessed by the open stars of the vertices. Example 6.4.3

Corollary 6 . 5 . 1 .

 651 (Pair vs quotient) Let (X, A) be an almost Euclidean compact pair. The pair C(X, A) has the surjection property if and only if C(X/A, ∅) = (C(X/A), X/A) has the surjection property.

Corollary 6 . 5 . 2 .

 652 Let (X, D) satisfy the homotopy extension property and A a compact subset of D. The following statements are equivalent: There exists a continuous function f : C(X) → X ∪ C(D) which is the identity on X ∪ C(A), There exists a continuous function f : C(X) → X ∪ C(D) which is the identity on X and satisfies f (C(A)) ⊆ C(D). Proof. The second condition precisely means that the inclusion map i : (X, A) → (X ∪ C(D), C(D))

Figure 6 . 2 :

 62 Figure 6.2: A pair (X, A) whose cone pair C(X, A) does not satisfy the surjection property

Figure 6 . 3 :

 63 Figure 6.3: A space X whose cone pair (C(X), X) does not satisfy the surjection property

Corollary 6 . 5 . 3 .

 653 (Cycles vs surjection property, implication) Let (X, A) be an almost Euclidean compact pair. If every regular cell C ⊆ X \A belongs to a relative cycle, then C(X, A) has the surjection property.

  q

Figure 6 . 5 :

 65 Figure 6.5: Illustration of the proof of (b) in Theorem 6.6.1

( 1 )

 1 ⇒ (3) is Corollary 4.4.3. (4) ⇔ (5) is Theorem 6.4.1. It remains to prove that (2) ⇒ (4) and (4) ⇒ (1) to obtain all the equivalences.

  (a) Star with 5 branches (b) Star with 1 branch

Figure 6 . 7 :

 67 Figure 6.7: The star pairs (X, A) have the surjection property (Example 6.7.1) (X in yellow, A in black)

Figure 6 . 8 :

 68 Figure 6.8: A union of 5 squares is the cone of a graph; the tip is at the center, the graph is in black (Example 6.7.2).

Figure 6 . 10 :

 610 Figure 6.10: Bing's house with two rooms and a half-cut of it (the full house is obtained by adding the symmetric reflection of the half-cut through the front vertical plane). It consists of two rooms, each of which can be accessed from outside through a tunnel crossing the other room. Each tunnel is linked by an internal wall to a side wall.

Figure 6 . 11 :

 611 Figure 6.11: The local cones in Bing's house: their bases (in black) are graphs that are unions of cycles. Each point of Bing's house is the tip of one of these three cones: two points are tips of the third cone, all the other points on the dashed lines are tips of the second cone, all the other points are tips of the first cone.

Corollary 6 . 7 . 6 .

 676 If (M, ∂M ) is a compact manifold with possibly empty boundary, then the pair C(M, ∂M ) has the surjection property, and hence it has computable type.Recall that this result is proved differently in Section 5.4, see Corollary 5.4.1.

  Computable Type and the Hilbert Cube . . . . . . . . . . . . . . . . . . . 2 1.1.2 Strong Computable Type . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Relative Strong Computable Type . . . . . . . . . . . . . . . . . . . . 2 1.1.3 Computable Type vs Topological Invariants . . . . . . . . . . . . . . . . . Invariants and Strong Computable Type . . . . . . . . . . . . . . 3 2) The invariants E n and H n . . . . . . . . . . . . . . . . . . . . . . . . 3 3) Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4) Uniformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.4 The -Surjection Property . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 A Necessary Condition to have Strong Computable Type . . . . . . . 5 1.1.5 Failing to have Computable Type . . . . . . . . . . . . . . . . . . . . . . . 5 1) Computable Witnesses . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2) Spaces without proper copies of themselves . . . . . . . . . . . . . . . 6 1.2 Descriptive Complexity of Topological Invariants . . . . . . . . . . . . . . Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.3 Separating Finite Topological Graphs . . . . . . . . . . . . . . . . . . . . 7 1.3 The Surjection Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characterization for Finite Simplicial Complexes . . . . . . . . . . 11 2) Finite 2-Dimensional Simplicial Complexes . . . . . . . . . . . . . . . 11 3) Boundaries and the Odd Subcomplex . . . . . . . . . . . . . . . . . . 11 4) Computable Type is Not Preserved by Products . . . . . . . . . . . . 12 1.1 General Results about (Strong) Computable Type 1.1.1 Computable Type and the Hilbert Cube Two notions of computable type have been introduced in the literature, we prove that they are equivalent.

		3
	1) Σ 0 2 6
	1.2.1 Characterization of the Π 0 1 Topological Invariants . . . . . . . . . . . . . .	6
	1.2.2 The Σ 0 2 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	6
	1) Strong Approximation and Σ 0 2 Invariants . . . . . . . . . . . . . . . .	6
	2)

Contents 1.1 General Results about (Strong) Computable Type . . . . . . . . . . . . . 2 1.1.1 8 1.3.1 The ( -)Surjection Property and Unions . . . . . . . . . . . . . . . . . . . 8 1.3.2 Finitely Conical Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.3 The Surjection Property for Cone Pairs . . . . . . . . . . . . . . . . . . . 9 1) Quotients and Retractions to Spheres . . . . . . . . . . . . . . . . . . 9 2) Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1 1.3.4 Applications to Computable Type . . . . . . . . . . . . . . . . . . . . . . 11 1) A

  is Euclidean if it is n-Euclidean for some n. We introduce the following class of spaces.

	Definition. (Definitions 6.2.1 and 6.2.2)
	A space is almost Euclidean if the set of Euclidean points is dense. A space is almost
	n-Euclidean if the set of n-Euclidean points is dense. A pair (X, A) is almost Euclidean
	(resp. almost n-Euclidean) if X \ A is almost Euclidean (resp. almost n-Euclidean).

  Effective countably-based T 0 -spaces . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.1 Classes of Computable T 0 -spaces . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.2 The Hilbert Cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Computable Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 Vietoris Topology, Topological Invariant and Descriptive Complexity . . 22 2.3.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4 Weihrauch reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

	I Chapter 2
	Preliminaries Background on Computable Topology
	Contents
	2.1
	13

  . Computably overt if it is closed and the set {i ∈ N : A ∩ B i = ∅} is c.e.,

	3. Computable if it is effectively compact and computably overt,
	4. Effectively open or Σ 0 1 if there exists a c.e. set E ⊆ N such that A = i∈E B i ,
	5. Effectively closed or Π 0 1 if its complement is effectively open.

  the image of A by the homeomorphism is B. A compact pair (Y, B) is semicomputable if Y and B are semicomputable. Now, we recall the Definition of computable type introduced by Iljazović. Definition 2.2.2. A compact metrizable space X has computable type if for every copy Y of X in the Hilbert cube, if Y is semicomputable then Y is computable.

A compact pair (X, A) has computable type if for every copy (Y, B) of (X, A) in the Hilbert cube, if (Y, B) is semicomputable then Y is computable.

  2.2. Definition 4.3.1. A compact metrizable space X has strong computable type if for every oracle O and every copy Y of X in the Hilbert cube, if Y is semicomputable relative to O, then Y is computable relative to O. A compact pair (X, A) has strong computable type if for every oracle O and every copy

  .2 Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.6.3 Wedge Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.7 Separating Finite Topological Graphs . . . . . . . . . . . . . . . . . . . . . . 90 5.7.1 Modulus of Local Connectedness . . . . . . . . . . . . . . . . . . . . . . . 90 5.7.2 Effective vs Non-Effective Classes . . . . . . . . . . . . . . . . . . . . . . . 91 5.7.3 Separating Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

  Corollary 5.2.1 implies that E n is a Σ 0 2 invariant, therefore every E n -minimal compact pair has strong computable type, by applying Theorem 4.3.2.

Definition 5.3.2. (The invariant E n ) For n ∈ N, a compact pair (X, A) is in E n if there exists a continuous function f : A → S n that has no continuous extension F : X → S n .

  Indeed, the dimension assumption cannot be dropped in general. The pair (B 4 , S 3 ) is in E 2 as witnessed by the Hopf map, but it is not E 2 -minimal, as (B 3 , S 2 ) (B 4 , S 3 ) is also in E 2 . However, if S 3 ⊆ Y B 4 then S 3 is a retract of Y so (Y, S 3 ) / ∈ E 2 . We now apply this technique.

	Proposition 5.3.6

n by assumption. Therefore, (Y, B) / ∈ E n by Lemma 5.3.1.

The statement is reminiscent of the discussion about minimality at the end of Section 4.3.2. However, E n is not an upper set (indeed, (B n+1 , S n ) ∈ E n but (B n+1 , B n+1 ) / ∈ E n ) so Lemma 4.3.1 does not apply.

invariant is sufficient for a compact space to have strong computable type whereas satisfying thesurjection property is necessary. Furthermore, we studied the measure of the non-uniformity of the computation. Finally, we investigated how some spaces fail to have computable type.

The relation ⊆, which is the set {(X, X ) ∈ K 2 (Q) : X ⊆ X }, is Π 0 1 in the topology τ lowV ×τ upV .

Proof. One has X X iff there exists a rational ball B intersecting X and such that X is contained in Q \ B. This condition on the pair (X, X ) is an effective open set in the topology τ lowV × τ upV . In other words, inclusion is Π 0 1 in that topology.

Proof of Proposition 4.4.4. It is easy to see that P and ↑P have the same minimal elements. Now assume that P is Π 0 1 in τ 2 V and let us show that ↑P is Π 0 1 in τ 2 upV . One has (X, A) ∈ ↑P ⇐⇒ ∃(X , A ) ∈ K 2 (Q), (X , A ) ⊆ (X, A) and (X , A ) ∈ P.

Using Lemma 4.4.1 and the assumption that P is Π 0 1 in τ 2 V , the relation R := {(X, A, X , A ) : (X , A ) ⊆ (X, A) and (X , A ) ∈ P} is Π 0 1 in the topology τ 2 upV × τ 2 V . The space (K 2 (Q), τ 2 V ) is effectively compact, so the projection of R on its first two components, which is ↑P, is Π 0 1 in the topology τ 2 upV by Proposition 2.1.1. The second item is now easy to prove. If P is Σ 0 2 in τ 2 V , then P = n P n where P n are uniformly Π 0 1 in τ 2 V . Therefore, ↑P = n ↑P n and ↑P n are uniformly Π 0 1 in τ 2 upV by the first statement, so ↑P is Σ 0 2 in τ 2 upV .

Failing to have Computable Type

It is easy to prove that the line segment does not have computable type. Indeed, the line segment I = [0, 1], embedded in the simplest way as [0, 1] × {q} ⊆ Q where q = (0, 0, . . .), is computable. However, if A ⊆ N is the halting set (a non-computable c.e. set) and x A = n∈A 2 -n-1 , then [x A , 1] × {q} is a copy of I which is semicomputable but not computable.

Similar constructions can be made with disks, and more generally n-dimensional balls. For other spaces, it may not be easy to create a semicomputable copy which is not computable, in this section we obtain some results about this.

Computable Witnesses

The findings presented in [6] shed light on the connection between topology and computability, offering valuable insights that can be employed to distinguish various notions of computability in specific scenarios. The study delved into situations where two topologies imposed on the same space result in distinct sets of computable points. As a concrete application, we provide a means to construct a semicomputable yet non-computable copy of a given compact set, a task that would be arduous to accomplish directly (Theorem 4.5.2).

Comparing Computability in Three Topologies Definition 4.5.1. On a space X endowed with two effective countably-based topologies τ 1 , τ 2 , we say that τ 1 is effectively weaker than τ 2 if the basic τ 1 -open sets are effective τ 2 -open sets, uniformly. If C is a subset of X, then a topology on X induces a topology on C, obtained by intersecting the open sets with C. We say that τ and τ agree on C if they induce the same topology on C.

Let (X, τ ) be a computable Polish space and τ 1 , τ 2 be effective countably-based topologies such that τ 1 is effectively weaker than τ 2 and τ 2 is effectively weaker than τ . We want to build a τ 1computable point that is not τ 2 -computable.

Recall the definition of a generically weaker topology (Definition 3.8 in [6]).

Definition 4.5.2. We say that τ 1 is τ -generically weaker than τ 2 if every set C ⊆ X on which τ 1 and τ 2 agree is τ -meager.

"meager" can be equivalently replaced by "nowhere dense" (see Remark 3.2 in [6]). An effective version of being generically τ -weaker was introduced in [6] which led to Theorem 4.5.1 stated below. It is an effective version of Definition 4.5.2, i.e. τ 1 is τ -generically weaker that τ 2 if and only if it is effectively so, relative to some oracle see Section 3.4 in [6]. We recall the following result (Theorem 3.5 in [6]), which is easily proved thanks to an effective Baire category theorem (see Theorem 2.1 in [6]). Theorem 4.5.1. If τ 1 is effectively τ -generically weaker than τ 2 , then there exists x ∈ X that is τ 1 -computable but not τ 2 -computable. Moreover, such a point can be found in any τdense Π 0 2 (τ )-set.

Note that the proof utilizes the priority method with finite injury. However, thanks to the effective Baire category theorem (refer to Theorem 2.1 in [6]), the proof is significantly streamlined. Essentially, one needs to provide a strategy to counteract a Turing machine's attempt to τ 2 -compute x, and the effective Baire category theorem seamlessly handles the interplay between these strategies.

Application to Computable Type

In this section, we give an application of Theorem 4.5.1 to give a clear and complete proof of Theorem 4.5.2. Let us first introduce the relevant notions.

In a more intuitive manner, the following definition signifies that the pair does not effectively fulfill the generalized -surjection property for every > 0. Definition 4.5.4. Let (X, A) be a compact pair in Q. For > 0, say that δ > 0 is an -witness if there exists a continuous function f

Say that (X, A) has computable witnesses if there exists a computable function sending each rational > 0 to a rational -witness δ > 0.

Theorem 4.5.2. Let (X, A) ⊆ Q be a pair of semicomputable compact sets. If it has computable witnesses, then (X, A) does not have computable type, i.e. there exists a semicomputable copy of (X, A) such that the copy of X is not computable.

The choice of strict and non-strict inequalities is not important, but is convenient as it makes the set of functions having modulus µ a closed subset of the space C(X, Y ) of continuous functions from X to Y .

For instance, a Lipschitz function with Lipschitz constant L has a modulus of uniform continuity µ(n) = n + log 2 (L) . Definition 4.5.6. Let X ⊆ Q. A copy Y of X in Q is a simple copy of X if there exists a homeomorphism f : X → Y such that both f and f -1 have a computable modulus of uniform continuity.

Theorem 4.5.3. (Computability of simple copies) Let X ⊆ Q be a computable compact set that does not properly contain a copy of itself. Let Y be a simple copy of X. If Y is semicomputable then Y is computable.

In particular, if X is computable and does not properly contain a copy of itself, then there is no geometrical transformation (scaling, rotation, translation) yielding a semicomputable copy of X which is not computable, and more generally there is no bilipschitz transformation yielding such a copy.

We need the following result which is folklore, but does not seem to appear in the literature.

Lemma 4.5.2

Let X ⊆ Q be a computable compact set and let µ : N → N be computable. The set

Proof. Let (x i ) i∈N be a dense computable sequence in X. If µ is a computable modulus of continuity of f , then an access to a name of f ∈ K µ is computably equivalent to an access to the values of f on this dense sequence, because using those values and the modulus of continuity, one can computably evaluate f at any point. In other words, the set K µ is computably homeomorphic to the set

therefore L µ is effectively compact, and so is K µ . We now prove the result. Proof of Theorem 4.5.3.

Let µ : N → N be a computable function. The set J µ of injective continuous functions f : X → Q such that µ is a modulus of uniform continuity for both f and

as the latter set is Π 0 1 , its intersection with K µ is effectively compact.

As J µ is effectively compact, its image P µ by φ is effectively compact as well, so P µ is a Π 0 1 -subset of K(Q) in the topology τ V . As X does not properly contain a copy of itself, any copy of X in P µ is P µ -minimal. Therefore, any semicomputable copy of X in P µ is computable, by the same argument as in the proof of Theorem 4.3.1.

Chapter 5

Descriptive Complexity of Topological Invariants

Proof. We assume that (X, A) / ∈ E(Y ) and show that (X , A

Converging Sequences

Now we give results showing how E(Y ) behaves w.r.t. converging sequences. First, if a sequence of compact pairs (X i , A i ) converges to a compact pair (X, A) in the Vietoris topology (i.e., if X i converges to X and A i converges to A), then whether all (X i , A i ) are (resp. are not) in E(Y ) does not imply that (X, A) is (resp. is not) in E(Y ). In other words, E(Y ) is neither closed nor open in the Vietoris topology.

Therefore in order to obtain results, we need more complicated assumptions on the way a sequence of compact pairs converges to a compact pair.

The first result can be used to prove that a compact pair is not in E(Y ) by approximating it with compact pairs that are not in E(Y ) in some way. It is taken from the definition of pseudo-cubes in [START_REF] Horvat | Computability of pseudo-cubes[END_REF]. Proposition 5.3.2 Let (X, A) be a compact pair and (X i , A i ) i∈N be a sequence of compact pairs contained in X such that 1. For every i, X i \ A i is open in X and is contained in (X \ A), 2. For every > 0, there exists i such that X \ (X i \ A i ) ⊆ N (A).

Proof. We assume that (X i , A i ) / ∈ E(Y ) for all i and show that (X, A) / ∈ E(Y ). Let f : A → Y be continuous. Y is an ANR, so there exists some continuous extension f of f to N (A), for some > 0 by Fact 3.3.1. Let i be such that X \ (X i \ A i ) ⊆ N (A) by (2), so f | A i can be defined and has a continuous extension F i : X i → Y by assumption.

We define a continuous extension

Note that X i and X \ (X i \ A i ) are closed and their intersection is A i , where these two functions agree. Therefore, F is well-defined and continuous and extends f . As f was arbitrary, we have proved that (X, A) / ∈ E(Y ).

The second result can be used to prove that a compact pair is in E(Y ) by approximating it with compact pairs that are in E(Y ), in a particular way.

Definition 5.3.1. Let Y be an ANR and X i converge to X in the upper Vietoris topology. We say that functions f i :

There are usually many ways of extending f to a neighborhood of X. However, any two of them are homotopic on a smaller neighborhood, which implies that the definition does not depend on the choice of an extension of f , and is intrinsic to the functions f i , f . We now prove that S I. Let > 0. We build a continuous function f : S → R 2 such that d(f, id) < . Let k ∈ N be large so that a k :=

The function f satisfies d(f, id) ≤ a k < and f (S) is homeomorphic to I, so S I.

Cylinder

We show another application of Theorem 5.6.1.

Theorem 5.6.2. If X is compact connected and not a singleton, then X X × [0, 1].

Proof. The argument is illustrated in Figure 5.4.

We first build a sequence of continuous functions f n : X → [0, 1] such that for every n and every x ∈ X, f n (B(x, 1/n)) = [0, 1]. As X is compact, there exists a finite set F n ⊆ X such that for every x ∈ X, there exists y ∈ F n with d(x, y) < 1/2n. Let δ < 1/2n be such that d(y, y ) ≥ 2δ for all distinct y, y ∈ F n . The function f n (x) = min(d(x, F n )/δ, 1) satisfies the sought condition (for every x, {0, 1} have preimages by

We assume that X is already embedded in (0, 1) N and consider its copy X 0 = {0}×X. For > 0, we then define g n : X 0 → Q as g n (0, x) = ( f n (x), x). The sets g n (X 0 ) are -deformations of X 0 , converging to [0, ] × X, which is a copy of [0, 1] × X. For instance, every Σ 0 2 invariant satisfied by the circle is also satisfied by the (geometric) cylinder, and every Σ 0 2 invariant satisfied by the line segment is satisfied by the filled square. However, the cylinder and the filled square satisfy the Σ 0 2 invariant "having dimension at least 2" (Proposition 5.6.1), which is not satisfied by the circle and the line segment.

Wedge Sum

We give another application of Theorem 5.6.1, showing that Σ 0 2 invariants cannot separate certain spaces, at least in one direction.

Theorem 5.6.3. Let (X, x 0 ) and (Y, y 0 ) be pointed compact spaces. If Y is connected and x 0 is not isolated in X, then X X ∨ Y .

Proof. The idea of the proof is that for any > 0, one can deform a small neighborhood of x 0 and make it converge to a small copy of Y of size . Let us give the detailed argument.

Chapter 6

The Surjection Property and Computable Type Contents Definition 6.2.1. Let X be a topological space and n

Every CW-complex without isolated point is almost Euclidean and every n-manifold is almost n-Euclidean. Definition 6.2.2. Let X be a topological space and n ≥

) is the border of the cell and is denoted by bd(C). The set f (B n \ S n-1 ) is the corresponding open cell and is denoted by op(C). A regular cell is a regular n-cell for some n ≥ 1.

Note that bd(C) and op(C) always contain the topology boundary and the interior of C respectively, but do not always coincide with them. For instance, in a simplicial complex, a free face of a simplex is contained in its border but not in its boundary.

A point x ∈ X is n-Euclidean if and only if x belongs to op(C) for some n-cell C ⊆ X.

Note that we work with compact metrizable spaces only.

The ( -)Surjection Property

The surjection property and the -surjection property were defined in Chapter 4. We recall their definitions and develop techniques to prove or disprove these properties.

Definition 6.3.1. A pair (X, A) satisfies the surjection property, if every continuous function

Let (X, d) be a metric space and A ⊆ X a closed set. For > 0, (X, A) satisfies the -surjection property if every continuous function f :

The space X has the -surjection property if the pair (X, ∅) does.

It satisfies the generalized -surjection property if every continuous function of pairs f :

Of course, the surjection property implies the -surjection property for any > 0.

Example 6.3.1

For every n ∈ N, the (n + 1)-dimensional ball and its bounding n-dimensional sphere form a pair (B n+1 , S n ) that has the surjection property. It is a consequence of an equivalent formulation of Brouwer's fixed-point theorem (Corollary 2.15 in [START_REF] Hatcher | Algebraic Topology[END_REF]) that § n is not a retract of B n+1 , but is a retract of any proper subset of B n+1 containing S n .

Example 6.3.2

Let n ∈ N and let d be a compatible metric on S n . If is smaller than half the distance between every pair of antipodal points, then S n has the -surjection property. It is a consequence of Borsuk-Ulam's theorem.

Example 6.3.3

If A X is a retract of X, then the pair (X, A) does not have the surjection property, as witnessed by the retraction.

Although the -surjection property depends on the metric of the particular copy of a pair (X, A), quantifying over yields a topological invariant, i.e. a property of the pair that is satisfied either by all copies or by none of them. Proposition 6.3.1 Let (X, A) be a compact pair. Whether there exists > 0 such that (X, A) has the -surjection property does not depend on the choice of a compatible metric on X.

Proof. By universality of Q, it is sufficient to prove that the -surjection property does not depend on the copy of (X, A) in Q. If (Y, B) is a copy of (X, A), then let φ : X → Y be a homeomorphism such that φ(A) = B. By compactness of X, φ is uniformly continuous so given > 0, there exists δ > 0 such that if d(x, x ) < δ then d(φ(x), φ(x )) < . If (Y, B) has the -surjection property, then we show that (X, A) has the δ-surjection property. Let f : X → X be continuous, satisfying

When the pair is well-behaved, the condition f | A = id A can be replaced by the weaker condition f (A) ⊆ A. Lemma 6.3.1 Let (X, A) be a compact pair satisfying the homotopy extension property and assume that A is an ANR whose interior is empty. The following statements are equivalent:

1. There exists > 0 such that (X, A) has the -surjection property, 2. There exists > 0 such that (X, A) has the generalized -surjection property.

When A has non-empty interior, the result still holds if one replaces the surjectivity of f by X \ A ⊆ im(f ) in condition 2. Proof. Of course 2. implies 1., we prove the other direction. We assume that 2. does not hold and prove that 1. does not hold. Let > 0. Let δ < /2 be such that functions to A that are δclose are /2-homotopic (Lemma 3.3.4). Let f : (X, A) → (X, A) be a non-surjective function satisfying d(f, id X ) < δ. There is an /2-homotopy h t :

We then apply Lemma 3.3.1, which implies that there is an /2-homotopy H t :

For well-behaved pairs, the -surjection property of the product implies the δ-surjection property of the two pairs for some δ.

Let X, Y and A X, B Y be compact ANRs.

If (X, A)×(Y, B) has the -surjection property, then (X, A) and (Y, B) satisfy the δ-surjection for some δ > 0.

indeed, the product of ANRs is an ANR (Theorem IV.7.1 in [START_REF] Borsuk | Theory of retracts[END_REF]), so X × B, A × Y and their Now assume 0 / ∈ im(f ). Let Y be the union of im(f ) and the ray from 0 to f (0). We need to show that Y is a proper subset of C(X). Let a, b be two distinct points of X. As im(f ) is closed and does not contain 0, the rays from 0 to a and b are not contained in im(f ). f (0) belongs to at most one of them, so the other ray is still not contained in Y . Therefore, Y is a proper subset of C(X).

The ( -)Surjection Property and Unions

In certains cases, the ( -)surjection property for a pair (X, A) can be established by proving the ( -)surjection property for subpairs covering (X, A).

The first result holds for cone pairs and countable unions.

If every pair C(X i , A i ) has the surjection property, then C(X, A) has the surjection property.

Proof. Assume that C(X, A) does not have the surjection property. There exist n ∈ N and a regular n-cell C ⊆ X \ A such that the corresponding quotient map q C : (X, A) → (S n , s) is null-homotopic.

For each i, the topological boundary ∂X i of X i is nowhere dense in X, so i∈N ∂X i is meager. By the Baire category theorem, its complement is dense, in particular it intersects op(C). Let x belong to the intersection and let i be such that x ∈ X i . As x / ∈ ∂X i , x belongs to the interior of

it is null-homotopic. Therefore, its restriction to (X i , A i ) is null-homotopic, implying that C(X i , A i ) does not have the surjection property.

Under certain conditions, the -surjection property is preserved by taking finite unions. Theorem 6.3.2. Let (X, A) and (X i , A i ) i≤n be compact pairs such that X = i≤n X i and A = i≤n A i . Assume that each topological boundary ∂X i is a neighborhood retract in X. If every pair (X i , A i ) has the -surjection property for some > 0, then (X, A) has the δsurjection property for some δ > 0.

Proof. For each i, there is a neighborhood U i of X i and a retraction r i :

Indeed, let V i ⊆ X be a neighborhood of ∂X i and ρ i :

The definition is consistent because the intersection of X i and V i \ int X (X i ) is ∂X i , on which ρ i coincides with the identity. Therefore, r i is well-defined and continuous, and is indeed a retraction.

Let > 0 be such that each (X i , A i ) has the -surjection property. Let δ > 0 be such that for each i ≤ n, N (X i , δ) ⊆ U i and for x, y ∈ U i , d(x, y) < δ implies d(r i (x), r i (y)) < .

Assume the existence of a non-surjective continuous function f : X → X such that f | A = id A and d(f, id X ) < δ. There exists i ≤ n and x 0 ∈ int X (X i ) which is not in the image of f . Let

Both f and r i are the identity on A, so f i is the identity on A. By choice of δ, d(f i , id X i ) < . Finally, f i is not surjective, because x 0 is not in its image: the only preimage of x 0 ∈ int X (X i ) by r i is x 0 , which is not in the image of f . by Proposition 3.2.1. Note that each OC(L i,j ) is open in X. Moreover,

where the last equality holds by Proposition 3.2.2. We have proved that (X, A) is finitely conical.

We now show that if (X, A) is finitely conical, coming with (L i , N i ) i≤n , then C(X, A) is finitely conical, witnessed by the pairs C(L i , N i ) together which the pair (X, A).

The first component of the pair C(X, A) is the cone C(X), which is covered by two open sets OC(X) and X × (0, 1]. Its second component is X ∪ C(A), which is covered by the two open sets OC(A) and X ∪ (A × (0, 1]). All in all, we have C(X, A) = OC(X, A) ∪ ((X, A) × ((0, 1], {1})).

The first pair OC(X, A) is finitely conical, because it consists of one open cone pair.

The second pair (X, A) × ((0, 1], {1}) = (X, A) × OC({0}, ) is a product of two finitely conical pairs, so it is finitely conical by the first statement. It is witnessed by the pairs (L i , N i ) * ({0}, ) = C(L i , N i ). Therefore, C(X, A) is the union of two open subspaces which are both finitely conical, so it is finitely conical. Remark 6.4.2

By similar arguments, it can be proved that if the pairs (X 1 , A 1 ) and (X 2 , A 2 ) are finitely conical, witnessed by (L 1 i , N 1 i ) i≤m and (L 2 j , N 2 j ) j≤n respectively, then their join (X, A) * (Y, B) is finitely conical, witnessed by the pairs

), which means that (X, A) * (Y, B) is covered by the open cones of these pairs. We will not use this result.

The -Surjection Property for Finitely Conical Pairs

We show that the -surjection property of a finitely conical pair reduces to the surjection property of each local cone pair, assuming that the L i 's are ANRs. This result is particularly useful because it enables one to check a global property by inspecting the local cones independently of each other. Theorem 6.4.1. Let (X, A) be a finitely conical compact pair coming with (L i , N i ) i≤n , where each L i is an ANR. The following statements are equivalent:

1. (X, A) has the -surjection property for some > 0, 2. All the cone pairs C(L i , N i ) have the surjection property.

The implication 2. ⇒ 1. does not follow from Theorem 6.3.2, because in a finitely conical pair (X, A), one has A = i C(N i ) while applying Theorem 6.3.2 would require A = i L i ∪ C(N i ). Proof. 1. ⇒ 2. Assume that some C(L i , N i ) does not have the surjection property and let > 0. C(L i , N i ) does not have the -surjection property by Proposition 6.3.3, which is witnessed by a non-surjective continuous function f : C(L i ) → C(L i ). We assume that C(L i ) is embedded in X in such a way that its topological boundary in X is L i . We then extend f as the identity outside C(L i ), showing that (X, A) does not have the -surjection property. Note that the extension of f is indeed continuous because f is the identity on the boundary of C(L i ). Proposition 6.5.1

Let (X, D) be a compact pair satisfying the homotopy extension property and A a compact subset of D. The following statements are equivalent 1. There exists a continuous function f : C(X) → X ∪C(D) which is the identity on X ∪C(A), 2. The quotient map q : (X, A) → (X/D, p) is null-homotopic (p being the equivalence class of D).

Proof. 1. ⇒ 2. The function f is a null-homotopy for pairs of the inclusion map i X : (X, A) → (X ∪ C(D), C(A)). Consider the quotient map p : X ∪ C(D) → (X ∪ C(D))/C(D) and observe that (X ∪ C(D))/C(D) can be identified with X/D. The function p • i X : (X, A) → (X/D, p) is precisely q, and the null-homotopy of i X composed with p is a null-homotopy of q.

2. ⇒ 1. Now assume that q is null-homotopic. First, we show that the inclusion i X : X → X ∪ C(D) is null-homotopic. The obvious nullhomotopy of the inclusion i D : D → X ∪ C(D) can be extended to a homotopy

be the continuous map induced by F 1 , i.e. satisfying

be a homotopy between H 0 = q and the constant map

The null-homotopy K t can be seen as a function g : C(X) → X ∪ C(D) which is the identity on X. However, g is not the identity on C(A), but we show how to modify g.

We use the following notation: C(X) is the quotient of [0, 1] × X obtained by identifying all the points (1, x), so we can express any point of C(X) as the equivalence class of a pair (t, x)

With this notation, we can see how g is defined on C(A). Let τ be the tip of C(D). For x ∈ A and t ∈ [0, 1], one has

Observe that g is constant on the segment S = {[t, x] : t ∈ [1/2, 1], x ∈ A}, and the idea is to contract S to a point as follows.

Let D > max x∈X d(x, A) and h : C(X) → C(X) be the continuous function defined by

and represented on Figure 6.1.

First, h is surjective because for each x ∈ X, h sends the segment {[t, x] : t ∈ [0, 1]} to itself and fixes its endpoints, so h sends the segment onto itself. As C(X) is compact Hausdorff, h is a quotient map.

Counter-examples

We consider a family of spaces which enables us to find counter-examples, and show that certain assumptions in the previous results cannot be dropped.

A Family of Spaces

Let m, n be two natural numbers and let f : S m → S n be continuous. One can attach B m+1 to B n+1 along their boundaries using f . We obtain the space X f = B n+1 ∪ f B m+1 , which is the quotient of B m+1 B n+1 obtained by identifying x ∈ S m+1 = ∂B m+1 to f (x) ∈ S n = ∂B n . It is illustrated in Figure 6.4.

The space X f . The ball B n+1 is visualized as the cone C(S n ), the center of the ball is the tip of the cone.

We will see that the properties of X f we are interested in only depend on the homotopy class of f . Therefore, by the Simplicial Approximation Theorem (Theorem 2C.1 in [START_REF] Hatcher | Algebraic Topology[END_REF]), we can always assume that f is a simplicial map, implying that X f is a finite simplicial complex.

We will need to consider the suspension construct. We recall that the suspension of a space X is SX = X * S 0 . One has SS n = S n+1 . To a map f : X → Y is associated its suspension Sf : SX → SY , which is naturally defined from f . Note that for f : S m → S n , one has Sf : S m+1 → S n+1 .

Cycles vs the Surjection Property

We show that the equivalences stated in Corollaries 6.5.4 and 6.5.5 are no more valid when the assumptions about the dimension of the space and the dimension of the regular cells are dropped. For clarity, let us summarize what we have so far.

For any compact pair (X, A), if C ⊆ X \ A is a regular n-cell, then we have the following implications 1. ⇒ 2. ⇒ 3.:

1. There is a retraction r : X \ op(C) → bd(C) which is constant on A, 2. The quotient map q C : (X, A) → (S n , s) is null-homotopic, 3. C does not belong to a relative cycle.

When (X, A) is a finite simplicial pair and dim(X) = n, all these conditions are equivalent by Corollaries 6.5.4 and 6.5.5. However we show that if dim(X) > n, then both implications 2. ⇒ 1. and 3. ⇒ 2. fail in general (still assuming that (X, A) is a finite simplicial pair).

Note that B n+1 is a regular (n + 1)-cell in X f . We relate the conditions 1., 2. and 3. for the space X f and the regular (n + 1)-cell C = B n+1 in terms of the properties of f . Theorem 6.6.1. Let m, n ∈ N and f : S m → S n be continuous. The space X f = B n+1 ∪ f B m+1 enjoys the following properties: a) There exists a retraction r :

Moreover, (C(X f ), X f ) has the surjection property iff Sf is not null-homotopic.

Proof. (a) Note that S n ∪ f B m+1 is the quotient of B m+1 by the equivalence relation x ∼ y iff x = y, or x, y ∈ S m and f (x) = f (y). Therefore, there is a one-to-one correspondence between the continuous functions r : S n ∪ f B m+1 → S n and the continuous functions

is a homotopy equivalence. We show that the homotopy class of q : X f → S n+1 is sent to the homotopy class of Sf : S m+1 → S n+1 under the equivalence, which implies that q is null-homotopic if and only if Sf is. More precisely, we show that q ∼ Sf • p (these two maps are illustrated on Figures 6.5a and 6.5b).

The sphere S n+1 is a union of two balls B n+1 , called the lower and upper hemispheres, joined at the equator. Let j : B n+1 B m+1 → S n+1 send B n+1 homeomorphically to the lower hemisphere of S n+1 , and send B m+1 to the upper hemisphere of S n+1 by applying the map C(f

The space X f is a quotient of B m+1 B n+1 and the map j respects that quotient: if x, y ∈ S m = ∂B m+1 and z ∈ S n = ∂B n+1 are such that f (x) = f (y) = z, then j(x) = j(y) = j(z) is z, seen as a point of the equator of S n+1 .

Therefore, j induces a continuous function k : X f → S n+1 , illustrated on Figure 6.5c. The functions q and Sf • p can be factored using k as follows. Let U and L be the upper and lower halves of S n+1 . The quotient of S n+1 by U or L is again S n+1 . Let q U , q L : S n+1 → S n+1 be the corresponding quotient maps. One has q = q U • k and Sf • p = q L • k. One easily sees that q U ∼ q L , so q ∼ Sf • p.

(c) As B n+1 is contractible, X f is homotopy equivalent to X f /B n+1 , which is homeomorphic to S m+1 . Therefore, H n+1 (X f ; G) ∼ = H n+1 (S m+1 ; G) which is trivial as m = n. As a result, any homomorphism defined on H n+1 (X f ; G) is trivial.

We prove the last assertion. For a regular (m + 1)-cell C ⊆ X f , the corresponding quotient map q C : X f → S m+1 is never null-homotopic, because it is a homotopy equivalence. Therefore, (C(X f ), X f ) enjoys the surjection property if and only if Sf is not null-homotopic.

The first application is that in Corollary 6.5.4, one cannot drop the assumption that all the cells have the same dimension as the space. Example 6.6.1

Let h : S 3 → S 2 be the Hopf map. It is known that h and Sh are not null-homotopic (Corollary 4J.4 in [START_REF] Hatcher | Algebraic Topology[END_REF]). Therefore, in X h the quotient map is not null-homotopic and (C(X h ), X h ) enjoys the surjection property although no regular cell belongs to a cycle, so the implication 3. ⇒ 2. fails for X h .

A second application is that the dimension assumption in Corollary 6.5.5 is needed. Example 6.6.2

Let again h : S 3 → S 2 be the Hopf map. The function 2h (which is the concatenation of h let (X, A) = (C(X f ), X f ) and Y = SX f . Lemma 6.6.1 SX f is homeomorphic to X Sf .

Proof. Both spaces are the quotients of (B n+1 B m+1 ) × [0, 1] by the equivalence relation generated by: (y, t) ∼ (f (y), t) for y ∈ S m , (6.6) (x, 0) ∼ (x , 0) and (x, 1) ∼ (x , 1) for x, x ∈ B n+1 , (6.7) (y, 0) ∼ (y , 0) and (y, 1) ∼ (y , 1) for y, y ∈ B m+1 . (6.8)

Proof. Taking the quotient by (6.6) first yields X f × [0, 1], and then the quotient by (6.7), (6.8) yields SX f . Taking the quotient by (6.7), (6.8) first yields SB n+1 SB m+1 and then the quotient by (6.6) yields X Sf .

Therefore,

) enjoys the surjection property whereas (C(X Sf ), X Sf ) does not. For cone pairs, the surjection property is equivalent to the -surjection property for any > 0 by Proposition 6.

The space Y = SX f is finitely conical with two open cones OC(X f ), so Y has the -surjection property for some > 0. The product Y × S 1 = SX f × SS 0 is finitely conical with four open cones OC(X Sf ), so Y × S 1 does not satisfy the -surjection property for any > 0. In both case we use Theorem 6.4.1.

Iteration of the Suspension

The existence of the function f in the proof of Theorem 6.6.2 can be extracted from the literature on homotopy groups of spheres and the suspension homomorphism, as we explain now. Theorem 6.6.3. For (m, n) = (7, 3) or [START_REF] Djamel | Strong computable type. Computability[END_REF][START_REF] Djamel | Computability of finite simplicial complexes[END_REF], there exists f : S m → S n such that Sf is not null-homotopic but S 2 f is.

The results in the literature are usually expressed in terms of the reduced suspension Σ rather than the suspension S (we use the notation Σ as in [START_REF] Hatcher | Algebraic Topology[END_REF], while the other references use the notation E). We recall that ΣX is defined as the quotient of SX = X * S 0 by the segment {x 0 } * S 0 for some x 0 ∈ X. However, for a CW-complex X, ΣX and SX are homotopy equivalent (Example 0.10 in [START_REF] Hatcher | Algebraic Topology[END_REF]) and if X, Y are CW-complexes and f : X → Y is continuous, then Σf is null-homotopic if and only if SX is. More precisely, the quotient map ϕ X : SX → ΣX is a homotopy equivalence and one has Σf

We use the following homotopy groups of spheres, that can be found in Hatcher [START_REF] Hatcher | Algebraic Topology[END_REF] ( §4.1, p. 339):

Applications to Computable Type

Whitehead Product. The Whitehead product is a way of combining an element f ∈ π p (S n ) with an element g ∈ π q (S n ) into their product [f, g] ∈ π p+q-1 (S n ). The suspension of a Whitehead product is always null-homotopic (see Whitehead [START_REF] Whitehead | On products in homotopy groups[END_REF], Theorem 3.11, p. 470).

Theorem 6.6.4. If f ∈ π p (S n ) and g ∈ π q (S n ), then one has Σ[f, g] = 0.

The next result was proved by Toda [58] ( §5.ii, p. 79). Lemma 6.6.2

There exists f : S 7 → S 3 such that Σf is not null-homotopic but Σ 2 f is.

Proof. Let i 4 : S 4 → S 4 be the identity. Let h 2 : S 3 → S 2 be the Hopf map and

ii that there exists f :

for some particular a 3 : S 6 → S 3 .)

Suspension Homomorphism. We will use the fact that the suspension homomorphism Σ : π 8 (S 4 ) → π 9 (S 5 ) is surjective, which follows from the famous Freudenthal suspension theorem (see Whitehead [START_REF] Whitehead | On products in homotopy groups[END_REF] p. 468 for the statement). The next theorem can be found in [START_REF] Freudenthal | Über die Klassen der Sphärenabbildungen. I. Große Dimensionen[END_REF].

Theorem 6.6.5. The suspension homomorphism

There exists f : S 8 → S 4 such that Σf is not null-homotopic but Σ 2 f is.

Proof. As π 9 (S 5 ) ∼ = Z 2 is non-trivial, let g ∈ π 9 (S 5 ) be non-zero. As Σ : π 8 (S 4 ) → π 9 (S 5 ) is surjective, let f ∈ π 8 (S 4 ) be such that Σf = g. As π 10 (S 6 ) is trivial, Σ 2 f = 0. Alternatively, Whitehead [START_REF] Whitehead | A generalization of the Hopf invariant[END_REF] ( §9, p. 234) shows that there exists γ ∈ π 8 (S 4 ) such that Σγ = ±[i 5 , i 5 ] = 0, where i 5 : S 5 → S 5 is the identity, therefore Σ 2 γ = 0. (It is defined as γ = J(γ 3 ) for some γ 3 in [START_REF] Whitehead | A generalization of the Hopf invariant[END_REF]).

Applications to Computable Type

The results obtained in the present chapter have immediate applications to the computable type property.

A Characterization for Finite Simplicial Complexes

Iljazović has demonstrated that every finite (topological) graph (G, V 1 ), where V 1 represents the set of vertices of degree 1, has computable type [START_REF] Iljazović | Computability of graphs[END_REF].

In this section, our objective is to extend the study of computable type to the broader class of finite simplicial complexes. We achieve this by topologically characterizing the pairs that exhibit computable type, utilizing the ( -)surjection property and unifying our findings from this chapter.

Finite simplicial complexes serve as the higher-dimensional counterparts to finite graphs, constructed by connecting simplices along their faces. This class encompasses a wide range of compact It is a classical example, introduced by Zeeman [START_REF] Zeeman | On the dunce hat[END_REF], of a space that is contractible but not intuitively so. It is a 2-dimensional simplicial complex with no free edge, i.e. no edge that belongs to one triangle only. Proposition 6.7.1

The dunce hat does not have computable type.

Proof. First, it is possible to turn the dunce hat into a simplicial complex, so we can apply our results. The vertices of the triangle are identified to a point v, and we show that the cone pair at that point does not have the surjection property. Indeed, in Figure 6.9c one can see that the cone pair at v is C(L, ∅) = (C(L), L) where L is the graph consisting of two circles joined by a line segment.

We apply Corollary 6.7.4: L is a finite graph containing an edge which is neither in a cycle nor in a path from N to N (N is empty), therefore C(L, N ) does not have the surjection property and hence the dunce hat does not have computable type. If A is the identified edges of the triangle, then it can be proved, by analyzing its local cone pairs, that the pair (D, A) has computable type. In particular, the cone pair at v is C(L, N ) where N consists of the two endpoints of the middle interval, so L is the union of two circles and a line segment between two points of N , hence C(L, N ) has the surjection property by Corollary 6.7.4. Remark 6.7.4

It is proved in [START_REF] Čelar | Talk at Computability and Complexity in Analysis[END_REF] that for any compact pair (X, A) where A has empty interior, if the quotient space X/A has computable type then the pair (X, A) has computable type. It is also proved that the converse implication fails, the counter-example is given by the circle X and a subset A consisting of a converging sequence together with its limit. The pair (X, A) has computable type, simply because X itself has computable type. However, X/A is homeomorphic to the Hawaiian earring which does not have computable type. This quotient is not a finite simplicial complex.

We give another counter-example of a quotient space which is a finite simplicial complex. Let L = C 1 ∨ I ∨ C 2 , X be the cylinder of L and A the two bases of the cylinder. Inspecting the local cones one can show that (X, A) has computable type but X/A does not.

2) Bing's House, or the House With Two Rooms

All the known examples of sets having computable type are non-contractible (note that we are not considering pairs, but single sets), and one might conjecture that no contractible set has computable type. We give a counter-example, which is a famous space that was defined as a counter-example for other properties. It was invented by Bing [12] and is now called Bing's house, or the house with two rooms. The set is depicted in Figure 6.10, together with a half-cut to help visualizing it. It is an example of a space which is contractible but not intuitively so. It can be endowed with a simplicial

Conclusion

T he -surjection property was proved in Section 4.4.3 to be a necessary condition to have strong computable type. For cones, it reduces to the surjection property. In this chapter, we have developed several techniques to establish or refute these properties, applicable to classes of spaces including finite simplicial complexes and compact manifolds.

We have established precise relationships between the ( -)surjection property and the homotopy of certain quotient maps to spheres, which enables us to take advantage of results from homology and homotopy theory. We have given applications of these techniques. The first one is that the cone of a compact manifold has computable type. The second one is that any finite simplicial complex together with its odd boundary has computable type. The third and most important application is an answer to a question raised in [START_REF] Čelar | Computability of products of chainable continua[END_REF], showing that the computable type property is not preserved by taking products.

We give a characterization of finite simplicial pairs which have computable type. Namely, we prove that a finite simplicial complex has (strong) computable type iff it satisfies the -surjection property iff the star at each vertex satisfies the surjection property. This gives a visual tool to prove computable type for 2-dimensional finite simplicial complexes. As a concrete application, we prove that Bing's house has computable type whereas the dunce hat does not.

The reduction to homology implies that whether a finite simplicial complex which has dimension at most 4 has computable type is decidable. We gather some results from Bredon [START_REF] Bredon | Topology and Geometry[END_REF] and Hatcher [START_REF] Hatcher | Algebraic Topology[END_REF]. We do not recall the definitions of homology groups, but recall results from algebraic topology and then show how they imply Theorem 5.4.1.

List of Figures

If X is a topological space, A ⊆ X is a subset and G is an abelian group, then for each n ∈ N one can define the nth homology group H n (X, A; G), which is an abelian group. When A = ∅, we write

Homology of manifolds is very well understood. We will mainly use the fact that homology can detect compactness of manifolds (Corollary V I.7.12 (p. 346) in [START_REF] Bredon | Topology and Geometry[END_REF]), as follows.

Theorem A.1.1. Let M be a connected n-manifold and assume that G contains an element of order 2. One has H n (M ) 0 ⇐⇒ M is compact.

The next result is a generalization to pairs (Corollary V I.7.11 (p. 346) in [START_REF] Bredon | Topology and Geometry[END_REF]).

Theorem A.1.2. Let M be an n-manifold and assume that G contains an element of order 2.

If A ⊆ M is closed and connected, then

Note that Theorem A.1.1 is just a particular case of Theorem A.1.2, with A = M .

A.2 Proof of Theorem 5.4.1

These results imply that if (M, ∂M ) is a compact connected n-manifold with (possibly empty) boundary, then removing a point makes the nth homology group trivial, because compactness is lost. The argument is very classical but we did not find this particular statement in any reference textbook, so we include a proof. For x ∈ M , we write M x = M \ {x}. Proof. If ∂M is empty, then it is a direct consequence of Theorem A.1.1, because M x is a noncompact connected n-manifold. Let us now consider the case when ∂M is non-empty.

Appendix B Coefficients

In Definition 6.5.1 we used the coefficient group T to detect relative cycles. One could equivalently use the groups Z k = Z/kZ (with k ≥ 2). The group Z can also give information. Although the next result is probably folklore, we did not find a suitable reference and include a proof for completeness.

Proposition B.0.1

Let (X, A) be a compact pair and x ∈ X \ A be n-Euclidean. The following statements are equivalent:

The homomorphism H n (X, A; T) → H n (X, X \ {x}; T) is non-trivial,

The homomorphism H n (X, A; Z k ) → H n (X, X \ {x}; Z k ) is non-trivial for some k ≥ 2, and they hold if the homomorphism H n (X, A) → H n (X, X \ {x}) is non-trivial.

Proof. We use the universal coefficient theorem for homology, and the fact that it is functorial in the space and in the coefficient group. We will write f k 1 for G = Z k and f 1 for G = T, and similarly for the other homomorphisms. We first assume that f : H n (X, A) → H n (S n , s) ∼ = Z is non-trivial and show that f k 2 is non-trivial for some k ≥ 2. Let q ∈ H n (S n , s) be a non-zero element of the image of f . Let k ≥ 2 be an integer that does not divide q, and take G = Z k in B.1, so that (q, 1) is not zero in H n (S n , s) ⊗ G. As (q, 1) belongs to the image of

2 must be non-trivial. We now assume that f k 2 is non-trivial for some k ≥ 2 and show that f 2 is non-trivial. Note that j 2 : Z k → T is the inclusion so it is injective. Therefore, the non-triviality of f k 2 implies the non-triviality of f 2 .

Finally, we assume that f 2 is non-trivial and show that f k 2 is non-trivial for some k ≥ 2. Again, the inclusion Z k → T sending [m] to [m/k] induces the following commuting diagram: 0

H n (X, A) ⊗ Z k H n (X, A; Z k ) Tor(H n-1 (X, A), Z k ) 0 0 H n (X, A) ⊗ T H n (X, A; T) Tor(H n-1 (X, A), T) 0 Proof. Note that Tor(H n-1 (X, A), T) is the torsion subgroup of H n-1 (X, A). Therefore, g 2 (c) is a torsion element of H n-1 (X, A), so there exists k ≥ 2 such that g 2 (c) ∈ im(i 3 ). As g k 2 is surjective, g 2 (c) = i 3 • g k 2 (e) for some e ∈ H n (X, A; Z k ). One has g 2 • i 2 (e) = i 3 • g k 2 (e) = g 2 (c), so c -i 2 (e) ∈ ker(g 2 ) = im(g 1 ), therefore there exists d ∈ H n (X, A) ⊗ T such that c -i 2 (e) = g 1 (d). Now assume that f 2 is non-trivial, and let c ∈ H n (X, A; T) have a non-zero image under f 2 . One has c = g 1 (d) + i 2 (e) as in Claim B.0.1, so f 2 • g 1 (d) = 0 or f 2 • i 2 (e) = 0. If f 2 • g 1 (d) = 0, then f 1 is non-trivial by B.1 so f is non-trivial and we can apply the first argument of the proof, implying that f r 2 is non-trivial for some r ≥ 2. If f 2 • i 2 (e) = 0, then f k 2 is non-trivial by B.2. It can happen that the homomorphism is trivial with the coefficient group Z but not with some Z k . It happens for instance if X is an n-manifold that is not orientable, with k = 2.