Lamine Diop

Arnaud Giacometti

Béatrice Markhoff

Arnaud Soulet

TTProfiler: Computing Types and Terms Profiles of Assertional Knowledge Graphs

As more and more knowledge graphs (KG) are published in the Web, there is a need of tools for abstracting their content for their producers to verify their result, and for their consumers to use it. This implies showing the schema-level patterns instantiated in the graph, with the frequency with which they are instantiated. A profile represents this information. In this paper, we propose a new type of profile that we call TT profile, for Types and Terms profile. It shows the used Types and predicates, and also the used Terms because of their paramount importance in most of KGs, especially in the Cultural Heritage (CH) domain. We present an algorithm for building a TT profile from an online KG's assertional part, and we report on experiments performed over a set of CH KGs.

Introduction

It has become widespread in the Cultural Heritage (CH) field to generate Knowledge Graphs from legacy datasets, using one or more ontologies [START_REF] Bikakis | Editorial: Special Issue on Semantic Web for Cultural Heritage[END_REF]. A knowledge graph (KG) is a dataset in RDF, i.e. a set of (subject, predicate, object) triples. CH KGs contribute to the Linked Open Data(LOD) construction, publicly offering inter-linked and semantically defined datasets, which is supposed to boost knowledge discovery and efficient data-driven analytics at a world-wide scale. However, using LOD datasets for analysis requires a clear idea of their content and this is a long-standing difficulty. It is not enough to know which ontologies are used, it is necessary to know how they are used, i.e. which of their components serve in that particular dataset, and in what way.

In the last ten years several proposals raised for helping users knowing what contains a given KG, by extracting its predicates, the types of entities they link, and some basic statistics, like ABSTAT [START_REF] Spahiu | ABSTAT: Ontology-Driven Linked Data Summaries with Pattern Minimalization[END_REF] does. In the same way, our aim is to generate such an abstract image of the KG as it is queryable online, by default without reasoning. From a given KG, ABSTAT builds a set of (C, P, D) triples with statistics, where C and D are types and P is a predicate. Such triple is called Abstract Pattern (AP). Figure 1 (left) shows the four first APs returned by ABSTAT when asking for the predicate dbo:country on a 2016 dump of DBpedia in English, using its online tool 1 . The first AP indicates that there are 560, 532 RDF triples (last column) in this KG for which the predicate dbo:country relates a subject of type dbo:Location to an object of type dbo:Country, which informs us that we can query locations and their associated countries. Figure 1 (right) presents the Basic Graph Pattern (BGP) able to compute an ABSTAT AP representing its instances, with n its frequency (number of its instances in the KG). Edges labeled with "a" represent the predicate rdf:type.

ABSTAT returns thousands of APs just for the predicate dbo:country from this dataset, several of them representing the same facts in the KG. For instance, dbo:Location and schema:Place in Figure 1 are probably both types of the subjects of predicate dbo:country that have objects of type dbo:Country, since the two APs have exactly the same frequency (i.e., 560,532). In other words, if the BGP in Figure 2 (a) was instanciated in the KG, then ABSTAT would generate four APs (cartesian product of subject's and object's types), all with the same frequency n. For representing each fact in the KG with only one AP, we propose to deal with APs where predicates relate not just types but sets of types, as in Figure 2 (a). Moreover, to the best of our knowledge there is no tool that highlights not only the types of the subject and object of a predicate, but also the terms used for objects in the KG. It is one thing to indicate that there are instances of crm:E22_Man-made_Object in the graph, but the fact that it contains information about coins, or burials, or garments, is much more interesting and precise. In KGs that use the CIDOC Conceptual Reference Model2 (hereafter CIDOC), this information is carried by terms, more precisely by URIs described in some thesauri, such as http://nomisma.org/id/coin for instance. This is the case because, quoting [START_REF] Boeuf | Definition of the CIDOC Conceptual Reference Model[END_REF], "CIDOC defines and is restricted to the underlying semantics of database schemata and document structures used in cultural heritage and museum documentation in terms of a formal ontology. It does not define any of the terminology appearing typically as data in the respective data structures; however it foresees the characteristic relationships for its use." The type crm:E55_Type is a gateway to these controlled vocabularies, but it is not the only one, for example crm:E57_Material can also be detailed in an ad hoc nomenclature. This choice is in line with the use of databases in CH communities insofar as it organises in ontology the entities of the domain and their relationships, but not the descriptive values, i.e. most of the values in databases. In general, these ones are listed and described elsewhere in authority lists, for interoperability purposes. This means that CIDOC-based KGs generally employ various sets of terms, which provide at least as much meaning as the used CIDOC types and predicates. For taking this into account, we propose abstract patterns showing terms used in the graph, as illustrated in Figure 2 (b), where t i denotes the instances of the variable ?t and the edge labeled with "prefLabel" represents the situation where the variable ?o is instantiated by a term and ?t by a label of that term. Detecting this situation depends on how are implemented the vocabularies. In this paper we consider those implemented with SKOS3 .

To sum-up our contribution, we deal with KGs that are sets of assertional knowledge, whose intentional part is formally defined by existing RDFS or OWL ontologies, and who contain instances of SKOS concepts, defined in existing SKOS thesauri. Given that the ontologies and thesauri used are not necessarily accessible online for programs, we present and discuss a program called TTProfiler to build a set of Types and Terms (TT) APs that we call a profile, by querying its online SPARQL endpoint 4 . The rest of this paper is organised as follows: in Section 2, we provide definitions for TT APs and profiles. In Section 3 we present the TTProfiler algorithm. We report in Section 4 our uses of its implementation on various online CH KGs. We conclude in Section 5.

Definitions and Problem Formulation

We use Description Logics (DL) [START_REF]The Description Logic Handbook: Theory, Implementation, and Applications[END_REF] formal notations for defining our problem: we consider a Web KG as a knowledge base (KB) K, composed of the TBox T (names and assertions about concepts and roles, respectively called types and predicates in this paper) and the ABox A (assertions about individuals, called entities and facts). For instance DBpedia is a KB K = (T , A), one example of assertion in T is dbo:Artist dbo:Person, meaning that the type dbo:Artist is subsumed by the type dbo:Person, i.e. all artists are persons. T also includes assertions like ∃dbo:birthYear dbo:Person, meaning that the predicate dbo:birthYear is defined for persons. On the ABox side, dbo:Person(dbr:Michelle_Obama) declares that entity dbr:Michelle_Obama is a person and birthYear(dbr:Michelle_Obama, 1964) states the fact that Michelle Obama was born in 1964. Also, some persons are related via the predicate dct:subject to a SKOS concept, for instance we find in DBpedia Person(dbr:Ringo_Madlingozi), skos:Concept(Category:1964) and dct:subject(dbr:Ringo_Madlingozi, Category:1964).

Our aim is to give an abstract of the ABox content, which in the Web of data is in general far bigger than the TBox. We work with the asserted KG (or ABox), not with the version one gets by applying a reasoner. By default, SPARQL endpoints do not perform entailments. We put in evidence all the types and predicates appearing in the ABox, whatever the ontologies they belong to, so we do not limit ourselves to only one given ontology. We also want to highlight the SKOS concept instances appearing in the ABox. To do so we look for skos:Concept instances or subjects of skos:prefLabel, and we also look for declared prefixes that correspond to some known thesauri. We do not use the ontologies and thesauri in the algorithm presented in this paper. When publicly available, they can be used latter on, together with profiles, for completing them. There may be cases in which the TBox is limited to few ontologies that are consistent by themselves and semantically compatible with each other. In those rare cases, a reasoning step combining the TBox and ABox could also be performed before or during the profile generation. This is out of the scope of this paper, because we want our proposal to work on the online Web, which presents too much uncertainty about the quality of the ontologies used (they are not even guaranteed to be accessible online for programs). This is why we limit our scope to the ABox content without considering the TBox, leaving it to other services to process the ontologies and thesauri for enriching the profile information afterwards, when needed. The algorithm presented in this paper builds a profile of the given ABox A, that is composed of TT AP (Types and Terms abstract patterns), which are triples whose subjects and objects are sets, as defined in Definition 1. In [START_REF] Spahiu | ABSTAT: Ontology-Driven Linked Data Summaries with Pattern Minimalization[END_REF], APs are triples (C, P, D), where C and D are types and P a predicate: we call them basic APs. TT APs generalise basic APs in two ways: first, objects can be either types or terms (labels of instances of skos:Concept). Second, both subjects and objects are sets (either set of types or set of terms), as illustrated in Figure 2.

Definition 1 (TT Abstract Pattern). Given an ABox A, a TT abstract pattern of A is a triple (C, P, D) such that C is a set of types in A, P is a predicate in A, and D is either a set of types in A or a set of terms appearing in A. Here a term is a string literal, label of an instance of skos:Concept. A TT abstract pattern (C, P, D) represents a fact P (a, b) of A if:

the entity a is an instance of each type in C (i. This definition can be declined in various versions. For instance, the subject and object of an AP could be generalised to types not actually appearing in A but defined using T , as owl:Thing, rdfs:Literal and so-called minimal types used in [START_REF] Spahiu | ABSTAT: Ontology-Driven Linked Data Summaries with Pattern Minimalization[END_REF]. Also for instances of skos:Concept, one could use some definitions in their respective thesaurus. As already noticed, contrary to [START_REF] Spahiu | ABSTAT: Ontology-Driven Linked Data Summaries with Pattern Minimalization[END_REF] There are many ways to define interesting statistics of a KG's assertional part. We may consider the global number of assertions C(a) for each type C, the global number of assertions P (a, b) for each predicate P , the global number of assertions P (a, b) for each SKOS concept b appearing in A... In this paper, we deal with the frequency of a TT AP, that is how many facts of A it represents. We call weight the function that associates with (C, P, D) its frequency in A. Moreover, for the sake of drawing clearly the TT profile as a graph, we aim at grouping the sets of types in such a way that each type appears in only one set (or node). For instance if we have in a TT profile P the TT APs

A 1 = ({C 1 , C 2 }, P 1 , {C 3 })), A 2 = ({C 1 , C 3 }, P 2 , {C 4 }), A 3 = ({C 1 }, P 3 , {t 1 , t 2 }) and A 4 = ({C 3 }, P 3 , {t 1 , t 3 }), with ω(A 1 ) = 20, ω(A 2 ) = 18, ω(A 3 ) = 100
and ω(A 4 ) = 50, then we merge sets {C 1 , C 2 }, {C 3 }, {C 1 , C 3 } and {C 1 } into a maximal set {C 1 , C 2 , C 3 } and sets {t 1 , t 2 } and {t 1 , t 3 } into a maximal set {t 1 , t 2 , t 3 }, which gives the representation shown in Figure 3.

Searching for maximal sets is searching for the components of the graph formed by the profile's nodes (subjects and objects of TT APs), with an edge connecting two nodes if and only if there is a non-empty intersection between these two nodes. The union of component's nodes is a maximal set. Computing the components of a graph is generally done by a linear depth-first search, but in Algorithm 1 we incrementally compute the maximal sets ϕ during the TT profile building. For the profile visualisation, maximal nodes can be represented by one of their types or terms, and the others can be shown on demand. As shown in Figure 3, edges are annotated with the corresponding AP and its weight.

We can now state our problem as follows: Given the assertional part of a knowledge base K = (T , A), how to efficiently generate and visualise a TT profile of A?

3 TTProfiler Algorithm TTProfiler computes a TT profile of an ABox A following a three steps procedure: 1) basic abstract patterns and statistics recovery, 2) TT profile computing, and 3) TT profile visualisation structure building.

Step 1: Basic abstract patterns and statistics recovery. We recover all basic abstract patterns (C, P, D) with w, their frequency, i.e. the number of instances of (C, P, D) in A (line 1). An assertion P (a, b) in A is said to be an instance of the abstract basic pattern (C, P, D) if and only if a is of type C in A (i.e., C(a) ∈ A) and b is either of type D or a term of a thesaurus.

Step 2: Profile computing. To fit Definitions 1 and 2, for each predicate appearing in a basic abstract pattern we group all types that have common instances (lines 5-11), and we also group terms for subjects having the same type (lines 12-14). For this last case, we associate to the predicate a weight equals to the sum of the weights computed in Step 1. With the resulting weighted TT APs, each fact P(a,b) is represented by only one pattern. Each computed TT AP is added into the TT profile P (line 15). We also incrementally compute the set ϕ of maximal nodes, incorporating in it the nodes C and D (that are sets of types or terms) (line 16). The incorporation of a node in ϕ consists in grouping its elements with other nodes containing them, as explained in Section 2 (cf. the example illustrated in Figure 3).

Step 3: Profile Visualisation structure computing. In this last step, for each weighted TT abstract pattern we replace its subject and object by their corresponding maximal node in ϕ (lines 19-20) and we add the resulting triple to the Profile Visualisation structure P V .

Regarding the complexity, Step 1 consists in querying the KG, so it depends on the SPARQL endpoint and the network capacities; Step 2 is linear in the number of predicates and quadratic in the number of basic abstract patterns computed in Step 1; Step 3 is linear in the number of TT abstract patterns.

Experiments with Cultural Heritage KGs

TTProfiler, whose code is published in Github (see Section 1), is devised to apply to KGs that can be queried online via a SPARQL endpoint. This requires Step 3), on small graphs, less than 1, 000, 000 triples, the TT profile generation takes about 0.06 seconds. For 91, 000, 000 triples it takes 1.15 seconds. TTProfiler is implemented in Java using the Jena library to query the public SPARQL endpoints. It was run on Windows 10 with an Intel core i7 processor and 32 GB of RAM.

We design this program as part of a French CH project called SESAMES5 and we test it with the archaeological KGs grouped in OpenArchaeo6 . Those graphs are generated from legacy databases, based on a common model which is a small excerpt of the CIDOC and its extensions. Even with such a restricted ontology, all types and predicates do not have instances in all KGs, so the visual query tool that OpenArchaeo provides could be complemented by the display of TT profiles to show what can be asked. In addition, the producers of these graphs use the TT profiles to inspect the results of the KG automatic generation, which is based on mappings expressed with tools like Ontop and X3ML. KG producers exactly know which predicates, types and terms should appear in the TT profiles, and can therefore easily detect anomalies in their mappings.

Besides the KGs in OpenArchaeo, we looked for other graphs using the CIDOC CRM and offering a SPARQL API usable by an application. Of those found, many are not always online and many do not answer to counting SPARQL queries of Step 1. We present in Table 1 nine graphs that are currently7 capable of answering the required queries. Seven of them are from OpenArchaeo (Kition, Iceramm, Arsol, Epicherchell, Outagr, Rita, and Aerba) and are rather small, while the Smithsonian's8 and Doremus's9 graphs are of different designs, use English terminologies, and are much larger. Doremus contains multilingual labelsTable 1 shows the number of edges and nodes in KGs, the number of distinct types/terms appearing in P, the number of TT APs (i.e., |P|) and maximal nodes (i.e., |P V |). Although the set of basic abstract patterns is already a condensed representation of the original graph, it can be too large to be easily visualised, hence the grouping of types and terms and the use of the notion of maximal node, which allows us to display graphs with less nodes, as shown in the last two columns of Table 1. Figure 4 gives an example of the graph visualisation offered to end-users.

Table 2 shows that the nine KGs use the CIDOC and eventually one or more of its extensions (CRMsci, CRMarch, CRMba). Doremus uses the so-called Erlangen implementation of the CIDOC (denoted ecrm). In addition, these KGs use terms of thesaurus, the PACTOLS10 for OpenArchaeo KGs, and an internal vocabulary for Doremus. When thesauri are used, the number of the terms is far larger than that of types. Concerning the predicates instantiated in each KG, here again, all KGs use the CIDOC or its extensions. As can be noticed, CRMba is used by ArSol in OpenArchaeo for one type and not for any predicate: this is because the extensions are attached to CIDOC by subsumption links. So, one can use extension classes with predicates that are defined in CIDOC. In general both types and predicates of extensions are used in the tested KGs. We present in Figure 4 a visualisation of a small profile, that of Epicherchell. In this graph, nodes suffixed by et_al are sets of terms and colours denote namespaces (e.g. blue for CRMsci). A click on an edge, here the predicate P4_has_time-span, displays its subject and object. Node's content is also displayed on demand, for instance in the bottom of the figure one have selected the node autel_et_al. These terms are used to describe the usage of objects. The profile shows that not only the objects are characterised by the type E57_Material with the predicate P45_consists_of, but it also contains the set of terms that are used in the KG for each material, with the node albatre_et_al that appears as object of the same property.

Related Works and Conclusion

Compared to works aiming at discovering the schema of a graph, or extracting modules or constraints from KGs, our problem is much simpler. TT profiles can be seen as a special kind of KG summaries as described in the recent and comprehensive surveys on KG summarisation [START_REF] Cebiric | Summarizing Semantic Graphs: A survey[END_REF]. Its authors classify the existing summarisation techniques in four classes: (i) structural methods that consider the paths (quotient graphs) or the subgraphs (with high centrality) in the KG, (ii) pattern mining methods that discover patterns in the KG and use them for showing a synthesis of the graph, (iii) statistical methods, that extract from the graph quantitative measures or statistics, and (iv) hybrid methods that combine some of the previous ones. As explained in [START_REF] Cebiric | Summarizing Semantic Graphs: A survey[END_REF], the proposals can be distinguished by their inputs and outputs: some works consider only ontologies, some others exploit only instances, and hybrid approaches process both. Outputs also differ ranging from a graph (not necessarily an RDF graph) to a set of items (e.g., rules or queries). Finally, very few of these proposals can be used online or make the source code available. We found only one work [START_REF] Troullinou | Ontology understanding without tears: The summarization approach[END_REF] that have been applied in the CH field, a structural method focused on centrality of concepts in the ontology. At the time of writing, it is not online usable and we did not succeed to get the sources. The closest to our proposal is ABSTAT [START_REF] Spahiu | ABSTAT: Ontology-Driven Linked Data Summaries with Pattern Minimalization[END_REF] dealing with both data and schemas, with a set of abstract patterns as output, usable online. To the best of our knowledge, none of the existing summarisation proposals can be tested on SPARQL endpoints (without loading locally a complete KG dump). Also, none of them output information about the KG terms. With respect to ABSTAT, our proposal can be analysed with regard to Thomas Kuhn's six criteria for characterising successful improvements in scientific theories [START_REF] Kuhn | Objectivity, Value Judgment, and Theory Choice[END_REF]: Generality (the scope of the theory is increased), Simplicity (the theory is less complicated), Explanatory power (the theory gives increased meaning), Fruitfulness (the theory can potentially meet more currently unspecified requirements), Objectivity (the theory provides a more objective shared understanding of the world), and Precision (the theory gives a more precise picture of the world). Computing a TT profile does not require access to the intensional part of the KG (the ontologies), which is mandatory for ABSTAT and other proposals. Moreover, our proposal works by querying online SPARQL endpoints, relieving users of the burden of downloading the KG dump. So, TTProfiler provides a support similar to ABSTAT's one, but in less constrained settings, which can be considered to improve the Generality of the tool. Clustering types and terms according to their use in the KG increases the representativity correctness, or Objectivity. Explanatory criteria is hardly applicable for these two tools, but as a TT profile exhibit more information about the KG's content by showing not Fig. 4: Epicherchell's profile only the types and predicates, but also the terms of shared controlled vocabularies that are used, it provides more information about the KG content's meaning. This important point could also be considered to improve the Precision criteria. Regarding Fruitfulness, TT abstract pattern definition allows us to propose a new form of profile, and other ones could still be defined based on TT APs.

As already mentioned, TTProfiler is used by OpenArchaeo's KG producers who know which predicates, types and terms should appear in the TT profiles and can detect anomalies in their KG production process. TT profiles are therefore a support to the automatic generation of KGs. This interesting aspect is also cited in [START_REF] Cebiric | Summarizing Semantic Graphs: A survey[END_REF] for summaries' applications. It emerged during our experiments, regarding the use of terms from thesaurus in particular, it revealed several errors in the graphs generated for OpenArchaeo, which could then be corrected. For the consumers of KGs, TT profiles offer an abstract of the graph content in the same way as ABSTATS' outputs, but with the terms added. It is a well-established practice in humanities and digital libraries to create and use authority lists of terms, i.e. shared controlled vocabularies, and our experiments demonstrated how interesting it is for humanists users to explore terms in TT profile visualisations. But the usefulness of terms goes beyond humanities: categories are first class citizen in Wikipedia, of paramount importance for crowdsourcing stakeholders; folksonomies are also a well known and studied phenomenon. User communities tend to organise themselves to create lists of terms for their needs of descriptions. Being it in a scholarly and structured way as in natural sciences, humanities and libraries, or simply spontaneously like in the social web, the phenomenon must be taken into account when trying to give an idea about a knowledge graph's content.

A TT profile can be used for supporting humans in discoverins KG content to retrieve the information they want. We are currently designing a Web visualisation tool for interactively showing the KG profile, and we plan to provide also an API for applications to query the profile. We are also studying ways of completing the TT profile with information from the ontologies, by extracting the minimal parts of ontologies useful for the profile's types and predicates. Another need demonstrated with our experiments is to build summaries from the profile for huge KGs. We would like to design algorithms for building summaries as sets of k connected nodes, for a given number k.

Fig. 1 :

 1 Fig. 1: ABSTAT (left), Basic Graph Pattern and its corresponding AP (right).

Fig. 2 :

 2 Fig. 2: Types (a) and Terms (b) Abstract Patterns.

  e., C(a) ∈ A for C ∈ C), and the entity b is an instance of each type in D (i.e., D(b) ∈ A for D ∈ D) or the entity b is an instance of skos:Concept and its prefLabel is in D (i.e., skos:Concept(b) and skos:prefLabel(b, t) and t ∈ D).

Definition 3 (

 3 Weight of a TT abstract pattern). The weight of the TT abstract pattern (C, P, D), denoted ω((C, P, D)), is the function that associates with (C, P, D) its frequency in A. ω((C, P, D))= |{P (a, b), P (a, b) ∈ A and P (a, b) is represented by (C, P, D) according to Definition 1}|.

Algorithm 1

 1 TTProfiler: Types and Terms Profiler Input: The ABox A of a knowledge base K = (T , A) Output: The TT profile P of A and its visualisation structure P V //Step 1: basic abstract patterns extraction from A and statistics computation 1: Let R = {((C, P, D), w)/(∃P (a, b) ∈ A) ∧ C(a) ∈ A ∧ (D(b) ∈ A ∨ (skos : Concept(b) ∧ skos:pref Label(b, D)))} where w is the number of instances P (a, b) in A for (C, P, D) //Step 2: TT profile computing: grouping types and terms in sets 2: LetP = {P/(∃((C, P, D), w) ∈ R)} P is the set of predicates in R 3: P ← ∅, ϕ ← ∅ϕ is a set of maximal sets of types or terms 4: for (P ∈ P) do grouping types and terms by predicates 5:for ((C1, P, D1), w1) ∈ R do 6:C ← {C1}, D ← {D1}, w ← w1 7:for((C2, P, D2), w2) ∈ R ∧ ((C1 = C2) ∨ (D1 = D2)) do 8: if (C1 = C2) ∧ (D1 = D2) ∧ (∀P (a, b) ∈ A : C1(a) ∈ A ∧ C2(a) ∈ A) then 9: C ← C ∪ {C2}group the types of subjects 10:if(D1 = D2) ∧ (C1 = C2) ∧ (∀P (a, b) ∈ A : (D1(b) ∈ A ∧ D2(b) ∈ A) then 11: D ← D ∪ {D2} group the types of objects 12: if (isLabel(D1)) ∧ (isLabel(D2)) ∧ (C1 = C2) then 13: D ← D ∪ {D2} group the terms 14: w ← w + w2 15: P ← P ∪ {((C, P, D), w)} 16: ϕ ← add(C, ϕ), ϕ ← add(D, ϕ) //Step 3: Profile visualisation structure 17: P V ← ∅ 18: for ((C, P, D), w) ∈ P do 19: A ← maxNode(C, ϕ) 20: B ← maxNode(D, ϕ) 21: P V ← P V ∪ (A, ((C, P, D), w), B) 22: return ( P, P V ) // add(ϕ, C) returns the set of maximal nodes ϕ having incorporated C // isLabel(D) returns true if D is a term, label of a skos:Concept in A // maxNode(C, ϕ) returns the maximal node that contains C to carefully write the SPARQL queries in Step 1 because of fair use policies applied by public SPARQL endpoints. Moreover, as already said about the time complexity, Step 1 of computing a TT profile depends on the configurations of the SPARQL endpoint and the network capacities. Considering only the client side computation (Step 2 and

  , if a or b have

		(({C1, C2}, P1, {C3}), 20)
			(({C1}, P3, {t1, t2}), 100)
	{C4}	(({C1, C3}, P2, {C4}), 18)	{C1, C2, C3}	{t1, t2, t3}
			(({C3}, P3, {t1, t3}), 50)
		Fig. 3: Graph with maximal sets

several types asserted in A (whether or not linked in T by a subsumption) then by Definition 1 the fact P(a,b) is represented by only one AP. Also contrary to

[START_REF] Spahiu | ABSTAT: Ontology-Driven Linked Data Summaries with Pattern Minimalization[END_REF]

, a fact P (a, b) having no type asserted for a, or having neither a type asserted for b nor any clue allowing to know that b belongs to a thesaurus, does not raise any AP. Given the set of APs generated from an ABox A according to Definition 1, we can associate statistics with those patterns, leading to the following definition of a TT profile: Definition 2 (TT Profile). Given an ABox A, a TT profile P of A is a set of pairs ((C, P, D), S) such that (C, P, D) is a TT AP generated from A, and S is a statistic value describing (C, P, D).

Table 1 :

 1 Knowledge graphs and TT profiles

		statistics for A		statistics for TT profile
	A	nb triples nb nodes language nb types & terms nb AP nb nodes
	Aerba	3,318	1,695	fr	5	3	5
	Epicherchell	3,488	1,372	-	31	15	13
	Kition	26,773	9,165	fr	72	31	19
	Iceramm	32,687	9,325	-	13	21	13
	Rita	40,479	10,769	-	184	6	7
	Outagr	79,420	39,573	fr	8	8	8
	Arsol	670,757	21,2143	-	94	34	17
	Smithsonian 2,542,142	969,172	en	18	35	18
	Doremus	91,093,326 24,141,972	en	599	678	146

Table 2 :

 2 Types, Terms and Predicates in the TT profiles

				Number of Types			Number of Terms
	K	crm crmsci crmarch crmba ecrm doremus	PACTOLS	vocabulary
	Aerba	4	0	0	0	0	0		0	0
	Epicherchell 9	1	0	0	0	0		21	0
	Kition	12	1	1	0	0	0		57	0
	Iceramm	11	1	0	0	0	0		1	0
	Rita	5	0	0	0	0	0		178	0
	Outagr	6	1	0	0	0	0		1	0
	Arsol	11	1	1	1	0	0		79	0
	Smithsonian 17	0	0	0	0	0		0	0
	Doremus	0	0	0	0	15	40		0	459
						Number of Predicates
			K	crm crmsci crmarch crmba ecrm doremus
			Aerba	3	0	0	0	0	0
		Epicherchell 13	2	0	0	0	0
			Kition	26	3	1	0	0	0
		Iceramm	19	2	0	0	0	0
			Rita	5	0	0	0	0	0
			Outagr	7	1	0	0	0	0
			Arsol	29	3	1	0	0	0
		Smithsonian 27	0	0	0	0	0
		Doremus	0	0	0	0	112	101

cidoc-crm.org/

https://www.w3.org/TR/skos-reference/

TTProfiler's codes are available at https://github.com/DTTProfiler/DTTProfiler

http://anr-sesames.map.cnrs.fr/

http://openarchaeo.huma-num.fr/explorateur/home

June, 2021

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Elisha Falbel

Antonin Guilloux

Qingxue Wang

GEOMETRIC STRUCTURES AND CONFIGURATIONS OF FLAGS IN ORBITS OF REAL FORMS

This is an introduction and a survey on geometric structures modelled on closed orbits of real forms acting on spaces of flags. We focus on 3-manifolds and the flag space of all pairs of a point and a line containing it in P(C 3 ). It includes a description of general flag structures which are not necessarily flat and a combinatorial description of flat structures through configurations of flags in closed orbits of real forms. We also review volume and Chern-Simons invariants for those structures.

Introduction

Riemannian metrics on 3-manifolds have been studied for a long time. In particular, locally homogeneous metrics are associated to certain classes of manifolds, the most important of them being hyperbolic metrics, that is, those of constant negative curvature. The geometrization conjecture by Thurston, proved by Perelman states that each compact 3-manifold has a decomposition into pieces, glued through tori, such that each of them has a locally homogeneous metric of finite volume defined in its interior.

A natural question is whether 3-manifolds admit other geometric structures in such a way that locally homogeneous structures might describe all of them by an analogous topological decomposition. In this survey we consider one special class of structures, namely, flag structures. They are modeled on the space of flags in P (C 3 ), that is, the space of all pairs formed by a point in P (C 3 ) and a line containing it (see section 2).

The two classical flag structures are path geometry and CR (Cauchy-Riemann) structures. Both geometries were studied by Elie Cartan (see [C24, C]) who constructed adapted fiber bundles and connections whose null curvature examples are locally homogeneous geometric structures. The Riemannian condition of finite volume can be translated in both geometries by a precise description of the structure on the boundary. We review in section 3 a definition of flag structure ( [FV]) in the Q. Wang was supported by NSFC grant #11371092. general context of not necessarily flat structures. The advantage of the definition is that it includes both CR and path geometries.

It is not known which manifolds admit flat structures of either path geometry or CR geometry. Both geometries have an underlying contact structure and it is not known yet if there are any constraints on the contact structure in the case a flat flag structure is defined on the manifold.

A locally homogeneous geometry can be described as a (G, X)-structure. Here X is a model space where a Lie group G acts transitively. The flat path or CR structures are described as (G 0 , X 0 )-structures where G 0 is a real form of SL(3, C) and X 0 is the unique closed orbit of the subgroup G 0 on the space of flags SL(3, C)/B, where B is the Borel subgroup of upper triangular matrices (see section 2).

In a more general form we are asking which manifolds are modeled on closed orbits of real forms. In the three dimensional case there are two non-compact real forms SL(3, C) and SU(2, 1) but both flat geometries can be seen as null curvature models of a flag structure (see [FV] and section 3). More generally, one can prove that all the null curvature models are obtained locally as totally real embeddings into the flag space SL(3, C)/B (see [FV] and section 3).

Geometric structures modeled on real forms can be constructed through configurations of flags contained in the closed orbits (see section 6). To each vertex of a simplex of a triangulation we associate a flag and because of the compatibilities given by the gluing maps between the simplices we obtain a representation (the holonomy) of the fundamental group of the manifold into a real form of PSL(3,[START_REF] Cartan | Sur le groupe de la géométrie hypersphérique[END_REF]. In some cases one is able to obtain a well defined geometric structure whose holonomy is the holonomy of the representation but this is not always the case. The general case of triangulations decorated with flags in P (C 3 ) was treated in [BFG] and, in higher dimensions, in [START_REF] Garoufalidis | The complex volume of SL(n,C)-representations of 3-manifolds[END_REF][START_REF] Garoufalidis | Gluing equations for PGL(n, C)representations of 3-manifolds[END_REF][START_REF] Dimofte | K-decompositions and 3d gauge theories[END_REF][START_REF] Guilloux | Deformation of hyperbolic manifolds in PGL(n,C) and discreteness of the peripheral representations[END_REF]. Decorated triangulations in the case of surfaces were introduced in [START_REF] Fock | Moduli spaces of convex projective structures on surfaces[END_REF] to describe projective structures and the decorated triangulations in the three dimensional case can be seen as a generalisation of this work. Another direct source of decorated triangulations is the work of Thurston constructing hyperbolic 3-manifolds by geometrizing triangulations via ideal hyperbolic tetrahedra (see [T]). The development of effective computations (SnapPea) with hyperbolic manifolds and their invariants gave motivation to understand invariants in this more general case.

In the last section we review some invariants of these decorated triangulations and relate them to the Chern-Simons invariants defined with the help of connections on fiber bundles associated to flag structures. The invariants are also related to K-theory via the third homology of SL(3, C) and we recall the definition of the Bloch group B(F ) of a field F which describes it in section 4.

Section 5 should be seen as an introduction to the subject of how to obtain an invariant from a decorated triangulation. Here, we aim at invariance with respect to a change in the triangulation. It contains an elementary proof of the triangulation independency of an invariant (defined in the pre-Bloch group) based on Pascal's theorem in the complex projective plane.

We prove (see [START_REF] Falbel | Duality and invariants of representations of fundamental groups of 3-manifolds into PGL(3,C)[END_REF] and section 6) that the invariants of flat real flag structures have values in the subgroup of conjugate invariant elements B + (C) ⊂ B(C) and we give in Proposition 4.8 the relation of this group and B(R).

We thank C. Gorodski for pointing to us Kostant-Wolff closed orbit theorem 2.3 some years ago and J. M. Veloso for many discussions.

Real orbits in flag spaces

In this section we describe the models of real flag structures. They are orbits under real forms of complex Lie groups acting on flag spaces.

2.1. Flag manifolds. A flag manifolds is defined as a quotient G/P where G is a connected complex semisimple Lie group and P ⊂ G a parabolic subgroup (that is, a connected complex Lie subgroup containing a maximal solvable Lie subgroup B, also called a Borel subgroup). In this survey we will deal mainly with G = SL(n, C) and mostly for n = 3.

The first examples of flag manifolds are given by the following construction. Let

1 ≤ d 1 < • • • ≤ d m < n be a sequence of positive integers. Consider the flag manifold of type (d 1 , • • • , d m ): Flag d1•••dm = {(V 1 , • • • , V m ) | dim(V i ) = d i and V 1 ⊂ • • • ⊂ V m ⊂ C n }
where the inclusion V i ⊂ V i+1 is strict. The group SL(n, C) acts transitively on this space with isotropy a parabolic group. The case where dim(V i+1 ) = dim(V i ) + 1 for all i is called the complete flags space and in this case the parabolic group is the Borel group of all upper triangular matrices in SL(n, C).

Flag manifolds are compact projective algebraic manifolds and the natural fibration G/P → G/B is holomorphic and G-equivariant.

Example 2.1. Our most important example is the flag space Flag 12 of complete flags (that is, lines and planes containing them) in C 3 . The group SL(3, C) acts on the space of flags with isotropy B, the Borel group of upper triangular matrices. We can describe therefore the space of flags as the homogeneous space Flag 12 = SL(3, C)/B. The space of flags is equipped with two projections. One projects the line of a flag into P(C 3 ) on one hand and, on the other, the plane into P(C 3 * ) viewed as a kernel of a linear form.

Flag 12 $ $ z z P(C 3 ) P(C 3 * )
The two projections correspond to the projections into P(C 3 ) = SL(3, C)/P 1 and P(C 3 * ) = SL(3, C)/P 2 where P 1 and P 2 are two different parabolic subgroups which fix, respectively, the line and the plane of the flag fixed by the Borel subgroup.

The projections define a complex contact distribution on the tangent space of Flag 12 generated by the tangent spaces to each of the fibers. We let T 1 and T 2 be the tangent spaces to the fibers.

One can embed Flag 12 into P(C 3 ) × P(C 3 * ) as the set of pairs (z, l) ∈ P(C 3 ) × P(C 3 * ) satisfying the incidence relation l(z) = 0.

From the identification

Flag 12 = SL(3, C)/B SU(3)/T 2 ,
with T 2 being the 2-torus, one obtains the long exact sequence of the fibration

T 2 → SU(3) → Flag 12 : • • • π 3 (T 2 ) → π 3 (SU(3)) → π 3 (Flag) → π 2 (T 2 ) → π 2 (SU(3)) → π 2 (Flag 12 ) → π 1 (T 2 ) → π 1 (SU(3)) → π 1 (Flag) → π 0 (T 2 ) → • • • .
We then observe that π 1 (Flag 12 ) is trivial and

π 2 (Flag 12 ) = Z ⊕ Z. It is also clear that π 3 (Flag 12 ) π 3 (SU(3)) Z.
Remark that the generators of π 2 (Flag 12 ) are given by the fibers of the two canonical projections.

In general, a flag manifold is simply connected and by a theorem of Wang ([Wa])(see also [Ti]) every compact simply connected homogeneous manifold is a torus bundle over a flag space.

Example 2.2 (The Klein correspondence). In the case n = 4 there is a classical description of the flag manifolds. In particular, Flag 2 is best described using the Plücker embedding:

pl : Flag 2 → P(Λ 2 (C 4 ))
defined for a plane Z, W determined by two linearly independent vectors as

pl( Z, W ) = [Z ∧ W ].
Using the basis e i ∧ e j , i < j, and corresponding homogeneous coordinates z ij , the map pl is an embedding into a quadric in P (C 6 ) (the Klein quadric) defined as

Q 4 = {[z ij ] ∈ P(C 6 ) | z 12 z 34 -z 13 z 24 + z 14 z 23 = 0}.
One can embed Flag 123 as a degree two variety in

P(C 4 ) × Q 4 × P(C 4 * ). Indeed, a point p = [x 1 , x 2 , x 3 , x 4 ] ∈ P(C 4 ) is in the plane m ∈ Q 4 if and only if p ∧ m = 0.
On the other hand, an hyperplane defined by l = [y 1 , y 2 , y 3 , y 4 ] ∈ P(C 4 * ) contains m if and only if the inner product l m is null. This is equivalent to say that (l ∧ e * i )(m) = 0 for all i. The flag manifolds correspondences are gathered in the following diagram:

Flag 123 z z $ $ Flag 12 $ $ Flag 13 $ $ z z Flag 23 z z Flag 1 Flag 2 Flag 3 2.2. Real orbits. Let G 0 be a real form of G = SL n (C).
That is a Lie subgroup with Lie algebra g 0 such that the Lie algebra of G is the complexification g = g 0 ⊗ C. Denote by τ the complex conjugation so that g 0 are the fixed points of that involution.

The following is a fundamental description of closed orbits in a flag space under a real form. It is attributed in part to Kostant in [START_REF] Wolf | The action of a real semisimple group on a complex flag manifold. I. Orbit structure and holomorphic arc components[END_REF]. Theorem 2.3 ([Wol69,Theorems 3.3 and 3.6]). Let Flag = G/P be a complex flag space and G 0 ⊂ G be a real form.

(1) there exists a unique closed G 0 -orbit T in Flag;

(2) for any x ∈ Flag, the closed orbit T is contained in the closure of G 0 x and

dim R G 0 x ≥ dim R T ≥ dim C Flag
with equality if and only Flag is a variety defined over R. In this case T is the set of real points of the associated real flag variety.

Example 2.4. The first example is that of SL(2, C) acting on P 1 (C). There are two real forms up to equivalence. The compact form SU(2) acts transitively on the flag space, as is always the case for any compact real form ( [Ti]). For G 0 = SL(2, R) the unique closed orbit is P(R).

Example 2.5. Consider SL(3, C) acting on P 2 (C). There are three real forms, the non-compact ones being SL(3, R) and SU(2, 1) (matrices preserving a hermitian form with two positive and one negative eigenvalues). The closed orbit of SL(3, R) is P(R 3 ) and geometric structures on surfaces modeled on this orbit are called projective structures. On the other hand, consider the group SU(2, 1) ⊂ SL(3, C) preserving the Hermitian form z, w = w * Jz defined on C 3 by the matrix

J =   0 0 1 0 1 0 1 0 0  
and the following cones in C 3 :

V 0 = z ∈ C 3 -{0} : z, z = 0 , V -= z ∈ C 3 : z, z < 0 . Let π : C 3 \ {0} → P(C 3 ) be the canonical projection. Then H 2 C = π(V -)
is the complex hyperbolic space and its boundary is

S 3 = π(V 0 ) = {[x, y, z] ∈ CP 2 | xz + |y| 2 + z x = 0 }.
is the closed orbit. The group of biholomorphic transformations of H 2 C is then PU(2, 1), the projectivization of SU(2, 1). Three-dimensional manifolds with geometric structures modeled on the the closed orbit S 3 are called CR structures.

Example 2.6. Consider SL(3, C) acting on Flag 12 (C 3 ). Again, the real forms are SL(3, R) and SU(2, 1). The closed orbit by the action of SL(3, R) is the set of real flags consisting of one point in P(R 3 ) and a line in P(R 3 ) passing through that point.

An element x ∈ S 3 ∈ P(C 3 ) gives rise to a flag in Flag 12 (C 3 ) where the line corresponds to the unique complex line tangent to S 3 at x. More explicitly, the map

φ CR : S 3 → Flag 12 (C 3 ) x → (x, . , x )
is an equivariant diffeomorphism of S 3 with the closed orbit.

Real flag structures

This section is a review of [FV]. We suppose M is a real three dimensional manifold in the following. Real flag structures are geometric structures on real manifolds which are generalizations of locally homogeneous geometric structures modelled on orbits of real forms in a flag space. They are not locally homogeneous in general and they are analogues of riemannian manifolds viewed as an abstraction of submanifolds of euclidean space.

We restrict ourselves to geometric structures related to the flag space Flag 12 as defined in [FV]. In the next subsection we identify the structure using totally real embeddings in the flag space. One defines two complex one dimensional distributions T 1 , T 2 ⊂ T M C which correspond to T 1 and T 2 on the flag space SL(3, C)/B (see example 2.1).

A flag structure on a real 3-manifold M is a generalization of this definition to the case there is no known real immersion into the flag space. The locally homogeneous examples arise naturally from closed orbits in the flag space by the action of a non-compact real form of SL(3, C).

3.1.1. Spherical CR geometry and SU(2, 1). Spherical CR geometry is modeled on the sphere S 3 equipped with a natural PU(2, 1) action as in Example 2.5. Another approach to the CR structure on the sphere is to use the map φ CR : S 3 → Flag 12 defined in Example 2.6 (that is x → (x, . , x )) as a totally real embedding and then define a flag structure as above.

3.1.2. Real path geometry and SL(3, R). Flat path geometry is the geometry of real flags in R 3 . That is the geometry of the space of all couples [p, l] where p ∈ RP 2 and l is a real projective line containing p. The space of flags is identified to the quotient

SL(3, R)/B R
where B R is the Borel group of all real upper triangular matrices. The inclusion

φ R : SL(3, R)/B R → SL(3, C)/B
is clearly a totally real embedding.

3.2. Flag structures in dimension 3.

Definition 3.2. A flag structure on M is a choice of two one dimensional subbundles

T 1 and T 2 in T C such that T 1 ∩ T 2 = {0} and such that T 1 ⊕ T 2 is a contact distribution.
The condition that T 1 ⊕ T 2 be a contact distribution means that, locally, there exists a one form θ ∈ T * M ⊗ C such that ker θ = T 1 ⊕ T 2 and dθ ∧ θ is never zero.

This definition contains two special cases, namely,

• CR structures, which arise when T 2 = T 1 .

• Path geometries, which are defined when T 1 and T 2 are complexifications of real one dimensional sub-bundles of T M . Path geometries ([C24] and section 8.6 of [IL]) and CR structures ( [C], see also [J]) in three dimensional manifolds were studied by Cartan. He defined what is called today a Cartan connection (see 3.3) for those two structures. The CR case was generalized to higher dimensions in [Ta, CM]. The general definition interpolates between the two geometries.

3.3. Connection forms and curvatures. In Riemannian geometry, the orthogonal coframe bundle is a geometric object which allows to define a connection and curvature. More generally, a homogeneous space G/H where H ⊂ G is a Lie subgroup might be used as a flat model space for a geometry on a manifold M which is encoded in a H-principal bundle

Y → M.
Recall that the fundamental vertical fields on Y are defined by X * (y) = d dt t=0 ye tX where e tX is the one parameter group generated by X ∈ h. In Cartan's idea, the bundle Y should be viewed as a non-flat analogue of the H-principal bundle G → G/H. A Cartan connection π is then a g-valued 1-form defined on Y which imitates the Maurer-Cartan forms. More precisely the connection satisfies π(X * ) = X for all X ∈ h, and is an isomorphism of vector spaces at each point. Moreover, it has the same invariance property (with respect to the subgroup H) as the Maurer-Cartan forms on the group G, that is

R h * π = Ad(h -1 )π,
where R h (g) = g.h is right multiplication by h and Ad(h -1 ) is the adjoint by h -1 defined on the Lie algebra g.

The curvature of a Cartan connection is given by

Π = dπ + π ∧ π.
The fundamental observation related to this construction is that, if the curvature is null, then the bundle Y is obtained locally through a local map from M to G/H by pulling-back the bundle G → G/H. Cartan analyzed several geometries, among them conformal and projective geometries before considering CR structures and path geometry over three manifolds. He defined a Cartan connection for each of those geometries on an appropriate bundle. This appropriate bundle is not always a coframe bundle but in several cases it is a coframe bundle over a natural bundle over the manifold. It corresponds to certain natural fiber bundles over the homogeneous space model.

3.3.1. The CR Cartan connection. Define SU(2, 1) = { g ∈ SL(3, C) | ḡT Qg = Q } with Hermitian form Q given by Q =   0 0 i/2 0 1 0 -i/2 0 0   .
The group SU(2, 1) acts on C 3 on the left preserving the cone

{ z ∈ C 3 | zT Qz = 0 }.
The projectivization of this cone is S 3 ⊂ P C 3 as in 2.5.

The elements of the Lie algebra su(2, 1) are represented by the matrices

  u -2iȳ w x a y z 2ix -ū   where ia ∈ R, z, w ∈ R, x, y ∈ C, u ∈ C and u -ū = -a.
Observe that the Lie algebra g = su(2, 1) is graded:

g = g -2 ⊕ g -1 ⊕ g 0 ⊕ g 1 ⊕ g 2 where g -2 =      0 0 0 0 0 0 z 0 0      g -1 =      0 0 0 x 0 0 0 2ix 0      g 0 =      u 0 0 0 a 0 0 0 -ū      g 1 =      0 -2iȳ 0 0 0 y 0 0 0      g 2 =      0 0 w 0 0 0 0 0 0      Observe that g 0 = R ⊕ u(1),
where

u(1) =      -iq/2 0 0 0 iq 0 0 0 -iq/2      with q ∈ R and R =      r 0 0 0 0 0 0 0 -r      with r ∈ R. Define the subalgebras h = g 0 ⊕ g 1 ⊕ g 2 a = R ⊕ g 1 ⊕ g 2 b = u(1) ⊕ g 1 ⊕ g 2 n = g 1 ⊕ g 2
The isotropy of the action of SU (2, 1)/K on S 3 at the point [1, 0, 0] T is the group H = CU (1) N (whose Lie algebra is h), represented (up to K) by matrices of the form

(3.3.1)   a -2iā b a(s -ib b) 0 ā a b 0 0 ā-1   where s ∈ R, b ∈ C. N is the Heisenberg group represented by matrices   1 -2i b s -ib b 0 1 b 0 0 1   .
Define the following subgroups of H; A = R + N (whose Lie algebra is a), B = U (1) N (whose Lie algebra is b) and N (whose Lie algebra is n). The inclusions between these groups, and their corresponding quotient spaces, can be visualized in the following diagrams

{Id} N | | " " A " " B | | H SU(2, 1) S 3 × C * y y & & S 3 × S 1 & & S 3 × R + x x S 3
A geometrical interpretation of SU (2, 1)/N is obtained considering the canonical bundle of the sphere, that is the restriction of Λ 2 C 2 to the sphere, and deleting from it the zero section. This is a trivial C * bundle over the sphere (it has a non-zero section given by the restriction of dz 1 ∧ dz 2 to the sphere). We obtain

SU (2, 1)/N S 3 × C * , SU (2, 1)/A S 3 × S 1 and SU (2, 1)/B S 3 × R + .
The construction of the appropriate H-principal bundle Y over a CR manifold M , follows the diagram above. Although Y isn't a coframe bundle over M it can be obtained as a coframe bundle over an intermediate R + -bundle over a M . This is a particular instance of Cartan's technique of prolongation.

Y M × C * y y ' ' M × S 1 & & E M × R + w w M
We will not describe here the construction of Y and it is Cartan connection (see [J]). However, the bundle E isomorphic to M × R + which depends only on the existence of a co-oriented contact structure is constructed easily. Each contact plane on M is given locally by 1-forms such that, in different neighbourhoods, differ by a positive function. One considers then the R + -bundle E of those contact forms. The bundle Y → E is a B-bundle and on the bundle Y → M there exists a Cartan connection.

Proposition 3.3 (Cartan, [C]). On the H-bundle Y → M there exists a 1-form π : T Y → su(2, 1) satisfying: 1. π p : T p Y → su(2, 1) is an isomorphism 2. If X ∈ h and X * ∈ T Y is the vertical vector field canonically associated to X then π(X * ) = X. 3. If h ∈ H then (R h ) * π = Ad h -1 π 3.3.2.
The flag structure connection. The Lie algebra of SL(3, C) decomposes in the following direct sum of vector subspaces:

sl(3, C) = g -2 ⊕ g -1 ⊕ g 0 ⊕ g 1 ⊕ g 2 , where g -2 =      0 0 0 0 0 0 z 0 0      , g -1 =      0 0 0 x 0 0 0 y 0      , g 0 =      u 0 0 0 v 0 0 0 w      with u + v + w = 0, g 1 =      0 x 0 0 0 y 0 0 0      , g 2 =      0 0 z 0 0 0 0 0 0      .
One obtains the graded decomposition of sl(3, C), where b = g 0 ⊕ g 1 ⊕ g 2 is the space of upper triangular matrices with null trace. The tangent space of

SL(3, C)/B at [B] is identified to sl(3, C)/b = g -2 ⊕ g -1 .
The choice of the reference flag whose isotropy is B defines a decomposition of

g -1 = t 1 ⊕ t 2 , with g -2 = [t 1 , t 2 ], corresponding to the two parabolic subgroups with Lie algebras p 1 = t 1 ⊕ g 0 ⊕ g 1 ⊕ g 2 and p 2 = t 2 ⊕ g 0 ⊕ g 1 ⊕ g 2 .
One propagates the decomposition of the tangent space at [B] to the whole flag space by the action of SL(3, C) to obtain the two fields of complex vectors T 1 and T 2 .

We proceed to obtain a C * -bundle E over a real three manifold equipped with a flag structure. We consider the forms θ on T C such that ker θ = T 1 ⊕ T 2 . Define E to be the C * -bundle of all such forms. This bundle is trivial if and only if there exists a globally defined non-vanishing form θ.

As in the case of PU(2, 1) and SL(3, R) one can obtain a frame bundle Y over E. The bundle Y can also be fibered over the manifold M . The bundle Y → M is a principal bundle with structure group H = B/C * where B is the Borel group of upper triangular matrices.

Observe that, although M is a real manifold, each fiber is a complex space of dimension five which can be identified to the Borel subgroup H. The real dimension of Y is 13 which is dim R SL(3, C) -3. This dimension difference does not allow us to obtain a genuine Cartan connection but a slight generalization of the definition.

Proposition 3.4 ([FV]

). There exists on Y a 1-form π :

T Y → sl(3, C) satisfying: 0. π p : T p Y → sl(3, C) is injective. 1. If X ∈ h and X * ∈ T Y is the vertical vector field canonically associated to X then π(X * ) = X. 2. If h ∈ H then (R h ) * π = Ad h -1 π
Note that contrary to the usual definition of Cartan connection we don't impose that π p : T p Y → sl(3, C) be an isomorphism as the dimensions are different.

A local characterization of null curvature flag geometry is given in the following theorem. Recall definition 3.1 of a totally real embedding φ : M → SL(3, C)/B and its associated flag structure which correspond to φ * (T 1 ) = T 1 and φ * (T 2 ) = T 2 on the flag space SL(3, C)/B. Theorem 3.5 ( [FV]). A totally real immersion φ : N → SL(3, C)/B with induced flag structure on T M C given by T 1 and T 2 as above has adapted connection having null curvature. Conversely a flag structure whose adapted connection has zero curvature is obtained locally through a totally real embedding as above.

Observe that null curvature does not define a unique flag structure on a real manifold but instead decides whether is can be embedded as a totally real submanifold in flag space.

One can then define the holonomy of a flat flag structure. Indeed, for any loop starting at a point p ∈ M one can follow the immersion of M along the path and obtain an element of PSL(3, C) which will translate the original neighbourhood to the final embedding of it.

Bloch group

In this section, we will review the (pre-)Bloch group of a field F , the Rogers's dilogarithm function and the relations with Algebraic K-theory. We will mainly discuss the cases for F = R or F = C.

Basic definitions and properties.

Definition 4.1. Let F be a field. Let Z[F \ {0, 1}] be the free abelian group with generators

[x], x ∈ F \ {0, 1}. A 5-term relation is an element of Z[F \ {0, 1}] with the following form: (4.1.1) T (x, y) := [x] -[y] + [ y x ] -[ 1 -x -1 1 -y -1 ] + [ 1 -x 1 -y ], x = y ∈ F \ {0, 1}.
Let T be the subgroup of Z[F \ {0, 1}] generated by all the 5-term relations T (x, y), x = y ∈ F \ {0, 1}. We define the pre-Bloch group P(F ) of F to be the quotient group Z[F \ {0, 1}]/T .

Next we consider the tensor product F * ⊗ Z F * , where F * is the multiplicative group of F . Let AS be the subgroup of F * ⊗ Z F * generated by the subset {x ⊗ y + y ⊗ x|x, y ∈ F * }. We define 2 F * := (F * ⊗ Z F * )/AS. For x, y ∈ F * , we will denote by x ∧ y the image of x ⊗ y in 2 F * .

Define a homomorphism λ 0 : Z[F \{0, 1}] → 2 F * as follows: for each generator

[z], z ∈ F \ {0, 1}, λ 0 ([z]) = (1 -z) ∧ z.
Then we can check that λ 0 (T (x, y)) = 0. Hence λ 0 induces a homomorphism

λ : P(F ) → 2 F * . Definition 4.2. The Bloch group B(F ) of F is the kernel of the homomorphism λ. It is a subgroup of P(F ).
In this paper we will mainly consider the cases F = R or F = C. When F = C, by [Sah, Sus], both P(C) and B(C) are uniquely divisible groups, i.e. they are Q-vector spaces. In particular they are torsion-free. On the other hand, if F = R, then there exists non-trivial torsion. For example, for all x ∈ R-{0, 1}, the element

[x] + [1 -x] ∈ B(R)
does not depend on x and has order six ( [Sus]).

The complex conjugation gives rise to a natural involution:

τ : Z[C \ {0, 1}] → Z[C \ {0, 1}].
As τ preserves the 5-term relation (4.1.1), it induces an involution on the pre-Bloch group P(C) which we will also denote by τ . Set

P(C) + = {x ∈ P(C)|τ (x) = x} and P(C) -= {x ∈ P(C)|τ (x) = -x}.
They are the corresponding ±1-eigenspaces of τ . Then we see

P(C) = P(C) + ⊕ P(C) -.
Analogously,

B(C) = B(C) + + B(C) -,
where

B(C) + = {x ∈ B(C)|τ (x) = x} and B(C) -= {x ∈ B(C)|τ (x) = -x}. Clearly B(C) + (resp. B(C) -) is a subgroup of P(C) + (resp. P(C) -).
Definition 4.3. The Bloch-Wigner function is defined as follows,

D(x) = arg (1 -x) log |x| -Im( x 0 log (1 -t) dt t ), ∀x ∈ C \ {0, 1}.
It is well-defined and real analytic on C-{0, 1} and extends to a continuous function on CP 1 by defining D(0) = D(1) = D(∞) = 0. It is well-known that it satisfies the 5-term relation (4.1.1). Hence it gives rise to a well-defined homomorphism:

D : P(C) → R, given by D( k i=1 n i [x i ]) = k i=1 n i D(x i ).
Since D(z) = -D(z) for any z ∈ C, we obtain that

D(u) = 0, ∀ u ∈ P(C) + .
For the proofs of the above-mentioned properties of the dilogarithm function D, see [B1] Lecture 6. 4.2. Rogers's dilogarithm. This function was introduced and studied in [Ro].

Here we use a slightly modified version (See [PS, Sus, Lic]).

Definition 4.4. The Rogers's dilogarithm is defined as follows:

L(x) = ∞ n=1 x n n 2 + 1 2 log x log (1 -x) - π 2 6 , 0 < x < 1.
Note it is differentiable with the derivative

L (x) = - 1 2 { log(1 -x) x + log(x) 1 -x }, 0 < x < 1.
By continuity, we have

L(1) = 0, L(0) = - π 2 6 .
It is known that (see [Ro, Kir, Za])

(4.2.1) L( √ 5 -1 2 ) = π 2 10 - π 2 6 = - π 2 15 , and
(4.2.2) L(( √ 5 -1 2 ) 2 ) = L( 3 - √ 5 2 ) = π 2 15 - π 2 6 = - π 2 10
.

By [Sus], we have the group homomorphism r : P(R) → R/π 2 Z, defined as follows: for each generator [FS] that on the torsion part of B(R), r is injective. It is conjectured that r is injective on the whole B(R).

[x], x ∈ R \ {0, 1}, (4.2.3) r([x]) =      L(x) mod π 2 Z, if 0 < x < 1 -L( 1 x ) mod π 2 Z, if x > 1 (-π 2 6 + L( 1 1-x )) mod π 2 Z, if x < 0 It was proved in
Example 4.5. Let u = (

√ 5-1 2 ) 2 and v = ( √ 5+1
2 ) 2 . We can check that both [u] and

[v] belong to B(R), i.e. λ([u]) = λ([v]) = 0. Note uv = 1. By (4.2.2) and (4.2.3), we have (4.2.4) r([u]) = L(u) mod π 2 Z = - π 2 10 mod π 2 Z and (4.2.5) r([v]) = -L( 1 v ) mod π 2 Z = -L(u) mod π 2 Z = π 2 10 mod π 2 Z.
Since r is injective on the torsion part of B(R), we see that in B(R), [u] = [v] and they both have order 10.

4.3.

For z ∈ C \ {0, 1}, we will still denote by [z] the element it represents in P (C). By [DS, Sus], We have in

P(C), ∀ z ∈ F \ {0, 1}, (4.3.1) [z -1 ] = -[z]; (4.3.2) [1 -z] = -[z].
Let V be a 2-dimensional complex vector space. Given five distinct points l i ∈ P(V ), 1 ≤ i ≤ 5, we define an element

F(l 1 , • • • , l 5 ) ∈ Z[C \ {0, 1}] as follows: (4.3.3) F(l 1 , • • • , l 5 ) := 5 i=1 (-1) i-1 [X(l 1 , • • • , li , • • • , l 5 )],
where li means that l i is omitted and

X(l 1 , • • • , li , • • • , l 5
) is the cross ratio as in Definition 5.1. We can choose a basis of V and express in the homogeneous coordinates

l 1 = [1, 0], l 2 = [0, 1], l 3 = [1, 1], l 4 = [x, 1], l 5 = [y, 1].
Since the l i are distinct, we have x, y ∈ C \ {0, 1} and x = y. By the Definition 5.1, we get:

(4.3.4) F(l 1 , • • • , l 5 ) = [x] -[y] + [ y x ] -[ 1 -y 1 -x ] + [ 1 -y -1 1 -x -1 ].
Note that this is different from the 5-term relation T (x, y) as in (4.1.1).

Lemma 4.6. With the notations as above.

(1). We have the following identity in Z[C \ {0, 1}]:

F(l 1 , • • • , l 5 ) = T (y -1 , x -1 ) + ([x] + [x -1 ]) -([y] + [y -1 ]).
(2). In

P(C), [F(l 1 , • • • , l 5 )] = 0.
Proof. (1). By (4.1.1), we have

T (y -1 , x -1 ) = [y -1 ] -[x -1 ] + [ y x ] -[ 1 -y 1 -x ] + [ 1 -y -1 1 -x -1
]. Now by Equation (4.3.4), we obtain

F(l 1 , • • • , l 5 ) = T (y -1 , x -1 ) + ([x] + [x -1 ]) -([y] + [y -1 ])
(2). By (4.3.1), we know that for any t

∈ C \ {0, 1}, [t] + [t -1 ] = 0 in P(C).
Hence, it follows from part (1). 2 4.4. Relations with Algebraic K-theory. Let F be a field. We first recall the definition of Milnor K-groups of F which was introduced in [Mil]. Let F * = F \ {0} be the multiplicative group of F . Let T (F * ) be its graded tensor algebra over Z.

That is,

T (F * ) = n≥0 (F * ) ⊗n ,
where

(F * ) ⊗0 = Z, (F * ) ⊗1 = F * , for n ≥ 2, (F * ) ⊗n = F * ⊗ Z • • • ⊗ Z F * (n factors).
Let I be the homogeneous ideal of T (F * ) generated by the elements of the form a ⊗ (1 -a), a ∈ F \ {0, 1}.

Definition 4.7. The Milnor ring of F is the graded ring K M * = T (F * )/I. Its degree n part is called the n-th Milnor K-group of F , denoted by K M n (F ). By definition, since the generators of I are of degree 2, we have

K M 0 (F ) = Z, K M 1 (F ) = F * . For n ≥ 2, as abelian group, K M n (F ) = (F * ) ⊗n /I n
, where I n is the subgroup generated by the tensors of the form

a 1 ⊗ • • • ⊗ a n with a i + a i+1 = 0 for some i, 1 ≤ i ≤ n -1.
The definition of Quillen's algebraic K-groups K * (F ) of F is more involved (See [Qui]). They are defined as the homotopy groups of a certain topological space which is closely related to the classifying space of the infinite general linear group GL(F ) of F . Here GL(F ) is the direct limit of GL(n, F ), n ≥ 1 with respect to the homomorphisms d n : GL(n, F ) → GL(n + 1, F ), defined by

d n (A) = A 0 0 1 . It is well-known that K 1 (F ) ∼ = K M 1 (F ) = F * and K M 2 (F ) ∼ = K 2 (F ).
Moreover, there is a natural homomorphism from K M n (F ) to K n (F ) for each n. The cokernel of this map is called the group of indecomposable elements, denoted by K ind n (F ). The first non-trivial case is when n = 3.

For more information and details about the algebraic K-theory, besides the original papers [Qui, Mil], we recommend the books [Sri, We] and the references therein. In this paper, we will focus on K 3 . Now for n = 3 and F an infinite field, we have the following fundamental exact sequence, due to Suslin ([Sus, Theorem 5.2]):

(4.4.1) 0 ----→ Tor(µ(F ), µ(F )) ∼ ----→ K ind 3 (F ) ----→ B(F ) ----→ 0, where µ(F ) is the group of roots of unity in F . Tor(µ(F ), µ(F )) ∼ is the unique nontrivial extension of Tor(µ(F ), µ(F )) by Z/2. When F = C, Tor(µ(C), µ(C)) ∼ = Q/Z, we have the exact sequence: (4.4.2) 0 ----→ Q/Z ----→ K ind 3 (C) ----→ B(C) ----→ 0. When F = R, Tor(µ(R), µ(R)) ∼ = Z/4Z, we have the exact sequence (4.4.3) 0 ----→ Z/4Z ----→ K ind 3 (R) ----→ B(R) ----→ 0.
We also know that (See [Sah])

K ind 3 (C) H 3 (SL(2, C), Z), K ind 3 (R) H 3 (SL(2, R), Z)
, where the group homologies are the homologies of SL(2, C), and SL(2, R) viewed as discrete groups.

Complex conjugation induces an involution on H 3 (SL(2, C), Z), which will be denoted by c. Set [DWS, Sah], the natural homomorphism g : H 3 (SL(2, R), Z) → H 3 (SL(2, C), Z) is injective and its image is equal to H 3 (SL(2, C), Z) + .

H 3 (SL(2, C), Z) + = {u ∈ H 3 (SL(2, C), Z)|c(u) = u}; and H 3 (SL(2, C), Z) -= {u ∈ H 3 (SL(2, C), Z)|c(u) = -u}. By
Let f : B(R) → B(C) be the natural homomorphism induced by the inclusion of R into C. The following proposition is probably known to the experts, but we can't find a proof in the literature. We include a proof here.

Proposition 4.8. The image of f is B(C) + and the kernel of f is the torsion

subgroup of B(R). Hence B(R)/{torsion} ∼ = B(C) + , equivalently, B(R) ⊗ Q ∼ = B(C) + .
Proof. By [DS, Sah, Sus], we have the following commutative diagram:

H 3 (SL(2, R), Z) t R ----→ B(R) ----→ 0 g   f   H 3 (SL(2, C), Z) t C ----→ B(C) ----→ 0
Both kernels of t C and t R consist of torsion elements. g is the natural homomorphism induced by the inclusion of SL(2, R) into SL(2, C). The complex conjugation acts both on H 3 (SL(2, C), Z) and B (C). Let H 3 (SL(2, C), Z) + and B(C) + be the corresponding subgroups which are invariant under the action. For a ∈ H 3 (SL(2, C), Z) or a ∈ B(C), we will denote by a its image under the complex conjugation. It is known that t C is compatible with the complex conjugation, i.e. for a ∈ H 3 (SL(2, C), Z), we have

t C (a) = t C (a).
Let's first show that the image of f is B(C) + . By the definition, the image of f is contained in B(C) + . Suppose a ∈ B(C) + ,i.e. a = a. Since B(C) is a Q-vector space, there is a unique b ∈ B (C)

such that b = 1 2 a, i.e. 2b = a. Since a ∈ B(C) + , so is b. Since t C is surjective, there is a c ∈ H 3 (SL(2, C), Z) such that t C (c) = b. Let x = c + c, then x ∈ H 3 (SL(2, C), Z) + and t C (x) = t C (c) + t C (c) = b + b = 2b = a. Since g maps H 3 (SL(2, R), Z) onto H 3 (SL(2, C), Z) + , ∃y ∈ H 3 (SL(2, R), Z) such that g(y) = x. By the diagram chasing, we see that f (t R (y)) = a. Therefore the image of f is B(C) + . If a is a torsion of B(R), since B(C) is torsion-free, we see f (a) = 0. Conversely suppose a ∈ ker f . Since t R is surjective, ∃b ∈ H 3 (SL(2, R), Z) such that a = t R (b). By the commutative diagram, we obtain t C (g(b)) = 0. Since ker t C is torsion, g(b) is a torsion. Since g is injective, b is a torsion. Hence a = t R (b) is a torsion. 2 4.5. H 3 (SL(n, C), Z) and K 3 (C). Let F be a field. The natural imbedding a n : SL(n, F ) → SL(n + 1, F ) defined by a n (A) = A 0 0 1 induces the natural homomorphism a(i, n) : H i (SL(n, F ), Z) → H i (SL(n + 1, F ), Z)
for each i and n.

The following proposition was proved by Sah.

Proposition 4.9 (Sah, [Sah]). Let F = C or R.

(1). a(3, n) :

H 3 (SL(n, F ), Z) → H i (SL(n + 1, F ), Z) are isomorphisms for n ≥ 3 ; (2). a(3, 2) : H 3 (SL(2, F ), Z) → H 3 (SL(3, F ), Z) is injective.
Moreover, for F = C, we have for each n ≥ 3,

K 3 (C) H 3 (SL(n, C), Z), and
K 3 (C) H 3 (SL(2, C), Z) K M 3 (C) K ind 3 (C) K M 3 (C).

Configurations of flags and their invariants

Three manifolds are triangulated by tetrahedra and the change from one triangulation to another is done through 3-2 or 4-1 moves. For manifolds which are locally a flag manifold it is natural to think of the tetrahedra with vertices given by flags. This will be described in the next section.

In order to obtain an invariant of the manifold, defined in terms of a triangulation and vertices associated to flags, one should prove that it it is independent of the triangulation. In other terms, one should prove invariance under 2-3 or 4-1 moves.

A natural definition involves cross ratios of a configuration of four flags and the invariance property translates into a 5-term relation. We will first recall the definition of the cross ratio of 4 distinct points on a projective line and then review the invariant of four flags. Proposition 5.4. Let C be an irreducible conic in the projective plane P(V ). Fix 4 distinct points l i ∈ C, 1 ≤ i ≤ 4. For any two points p = q on C with p = l i , q = l i , 1 ≤ i ≤ 4, we have the equality:

X(p|pl 1 , pl 2 , pl 3 , pl 4 ) = X(q|ql 1 , ql 2 , ql 3 , ql 4 ),
where for two distinct points u, v ∈ P(V ), we denote by uv the line passing through them.

Proof. Since C is irreducible, pl 1 , pl 2 , pl 3 , pl 4 are four distinct lines passing through p and ql 1 , ql 2 , ql 3 , ql 4 are four distinct lines passing through q. Let a = l 1 l 2 ∩ pl 3 , b = ql 2 ∩ pl 4 , c = l 1 l 4 ∩ ql 3 . By Pascal's theorem (see [START_REF] Fulton | Algebraic curves. An introduction to algebraic geometry[END_REF]Section 5.6 Corollary 1] for a proof), a, b, c are collinear. Let d = l 1 l 2 ∩ pl 4 , e = l 1 l 4 ∩ ql 2 . By Proposition 5.3, we have

X(p|l 1 , l 2 , l 3 , l 4 ) = X(l 1 , l 2 , a, d) = X(b|bl 1 , ql 2 , ba, pl 4 ),

and

X(q|l 1 , l 2 , l 3 , l 4 ) = X(l 1 , e, c, l 4 ) = X(b|bl 1 , ql 2 , bc, pl 4 ).

Now since a, b, c are collinear, we have lines ba = bc. Therefore, X(b|bl 1 , ql 2 , ba, pl 4 ) = X(b|bl 1 , ql 2 , bc, pl 4 ).

The proposition follows. 2

Remark 5.5. See [START_REF] Kaplansky | Linear algebra and geometry[END_REF]Page 117], where the result is proved by direct computations using a suitable basis and it is used to prove Pascal's theorem.

5.2.

In this section, we will take V to be C 3 . A (complete) flag of V is an ascending sequence of linear subspaces of

V : {0} ⊂ V 1 ⊂ V 2 ⊂ V with dim V 1 = 1, dim V 2 = 2.
Let Flag(V ) be the set of all flags of V . For a positive integer m, we define

Flag m (V ) = {(f 1 , • • • , f m )|f i ∈ Flag(V )
, 1 ≤ i ≤ m} to be the set of m ordered flags of V . Note that P GL(3, C) acts naturally on Flag(V ), and it acts term-byterm on Flag m (V ). The configuration space of m ordered flags of V is defined as the orbit space of Flag m (V ) by the action of P GL(3, C). We will denote it by

Conf m . For (F 1 , • • • , F m ) ∈ Flag m (V ), we will denote by [(F 1 , • • • , F m )] its orbit in Conf m .
Since dim V = 3 and a hyperplane of V corresponds to the kernel of a linear functional, we obtain

Flag(V ) = {([x], [f ]) ∈ P(V ) × P(V * ) | f (x) = 0}.
Here

[x] (resp. [f ]) means the 1-dimensional subspace generated by x ∈ V (resp. f ∈ V * ). ([x], [f ]) with f (x) = 0 represents the flag: {0} ⊂ [x] ⊂ ker f ⊂ V . Definition 5.6. Let [([x 1 ], [f 1 ]), • • • , ([x m ], [f m ])] ∈ Conf m . We say it is a generic configuration if f i (x j ) = 0, ∀i = j and [x 1 ], • • • , [x m ]
are in generic position in P(V ) (i.e. no three of them lie on a line in P(V )). We will denote by Gc m the configuration space of m ordered generic flags.

In the following we will call a generic configuration in Gc 4 a generic tetrahedron of flags. Now following [BFG] we will introduce the coordinates for them. Let T = ([x i ], [f i ]) 1≤i≤4 be a generic tetrahedron of flags. There are 12 coordinates for the 12 oriented edges of the tetrahedron. The coordinate z ij (T ) associated to the edge ij, 1 ≤ i = j ≤ 4 is defined as follows. We first choose k and l such that the permutation (1, 2, 3, 4) → (i, j, k, l) is even. Then the set of all the lines in P(V ) passing through the point x i is a projective line. We have four distinct points on this projective line: the line ker f i and the three lines x i x l passing through x i and x l for l = i. We define z ij (T ) as the cross ratio of these four points:

z ij (T ) := X((ker f i ), (x i x j ), (x i x k ), (x i x l )).
By the basic properties of the cross ratio, we see that the edge coordinates leaving a vertex i are related by:

z ik (T ) = 1 1 -z ij (T ) z il (T ) = 1 - 1 z ij (T ) (5.2.1)
The following proposition is proved in [BFG]. Definition 5.8. Let T ∈ Gc 4 be a generic tetrahedron with edge coordinates z ij (T ), 1 ≤ i = j ≤ 4. We define a P(C)-valued invariant: T )] + [z 21 (T )] + [z 34 (T )] + [z 43 (T )] ∈ P (C).

β(T ) = [z 12 (
Let P = ([x i ], [f i ]) 1≤i≤5 ∈
Gc 5 be a configuration of of 5 ordered generic flags. Then we have 5 generic tetrahedra T i of flags, 1 ≤ i ≤ 5, where T i is the generic tetrahedron obtained from P by deleting the i-th flag ([x i ], [f i ]) of P . The order of the 4 flags of T i is induced by the order of P . Then we have our main theorem: Theorem 5.9. With the notions as above. We have the following identity in P(C):

5 i=1 (-1) i-1 β(T i ) = 0.
Proof. By definition, we have (T 3 )] + [z 21 (T 3 )] + [z 45 (T 3 )] + [z 54 (T 3 )], β(T 4 ) = [z 12 (T 4 )] + [z 21 (T 4 )] + [z 35 (T 4 )] + [z 53 (T 4 )], β(T 5 ) = [z 12 (T 5 )] + [z 21 (T 5 )] + [z 34 (T 5 )] + [z 43 (T 5 )]. To simplify the notations, we will denote by ij the line passing through the points [x i ] and [x j ]. Observe that for each point [x i ] ∈ P(V ) of P , (1 ≤ i ≤ 5) there are 5 distinct lines passing through [x i ], i.e. ker f i and lines ik, 1 ≤ k ≤ 5, k = i, and the order of the 5 lines is given as follows: for 1 ≤ j ≤ i -1, l j = ij; l i = ker f i ; for i + 1 ≤ j ≤ 5, l j = ij. Now by Lemma 4.6 and the definition of the invariants z ij (T ), we have the following five identities in P(C), one for each [x i ]: (12,13,14,15)] -[Z 13 (T 2 )] + [z 12 (T 3 )] -[z 12 (T 4 )] + [z 12 (T 5 )] = 0, (21,23,24,25)] + [z 21 (T 3 )] -[z 21 (T 4 )] + [z 21 (T 5 )] = 0, (31,32,34,35)] -[z 35 (T 4 )] + [z 34 (T 5 )] = 0, (41,42,43,45)] + [z 43 (T 5 )] = 0, (51,52,53,54)] = 0. Hence we have the following identity in P(C):

β(T 1 ) = [z 23 (T 1 )] + [z 32 (T 1 )] + [z 45 (T 1 )] + [z 54 (T 1 )], β(T 2 ) = [z 13 (T 2 )] + [z 31 (T 2 )] + [z 45 (T 2 )] + [z 54 (T 2 )], β(T 3 ) = [z 12
I 1 = [X
I 2 = [Z 23 (T 1 )] -[X
I 3 = [Z 32 (T 1 )] -[z 31 (T 2 )] + [X
I 4 = [Z 45 (T 1 )] -[z 45 (T 2 )] + [z 45 (T 3 )] -[X
I 5 = [Z 54 (T 1 )] -[z 54 (T 2 )] + [z 54 (T 3 )] -[z 53 (T 4 )] + [X
(5.2.2) 0 = (12,13,14,15)] -[X (21,23,24,25)] + [X (31,32,34,35)] (41,42,43,45)] + [X (51,52,53,54)].

5 i=1 I i = 5 i=1 (-1) i-1 β(T i ) + R, where R = [X
-[X
Since the five point [x i ] are in general position in the projective plane P(V ), there is a unique irreducible conic C passing through them. Pick a point q ∈ C such that q = [x i ], 1 ≤ i ≤ 5. By Proposition 5.4, every cross ratio appearing in R is equal to a cross ratio of lines passing through q. That is, X(12, 13, 14, 15) = X(q|q2, q3, q4, q5), X(21, 23, 24, 25) = X(q|q1, q3, q4, q5), X(31, 32, 34, 35) = X(q|q1, q2, q4, q5), X(41, 42, 43, 45) = X(q|q1, q2, q3, q5), X(51, 52, 53, 54) = X(q|q1, q2, q3, q4),

where qi means the line passing through the points q and [x i ]. Hence we get

R = 5 i=1 (-1) i-1 X(q|q1, • • • , qi, • • • , q5),
By Lemma 4.6, we obtain R = 0 in P(C). Therefore, by (5.2.2)

5 i=1 (-1) i-1 β(T i ) = 5 i=1 I i = 0.
2 Remark 5.10. For a 3-dimensional manifold M which can be triangulated by finitely many tetrahedra, there are interesting P(C)-valued invariants for representations of π 1 (M ) in P U (2, 1) and P GL(3, C) ( [START_REF] Falbel | A combinatorial invariant for Spherical CR structures[END_REF][START_REF] Bergeron | Tetrahedra of flags, volume and homology of SL(3)[END_REF]). In order to show that the invariants are well-defined, it is crucial to prove that the invariants are independent of the 2-3 moves of the triangulations of M . Theorem 5.9 is equivalent to this. The proof we provide here gives an intuitive and geometric explanation of the calculations carried out in [Fa].

Representations

In this section we recall the definition of decorated triangulations of 3-manifolds following [START_REF] Bergeron | Tetrahedra of flags, volume and homology of SL(3)[END_REF][START_REF] Garoufalidis | Gluing equations for PGL(n, C)representations of 3-manifolds[END_REF][START_REF] Dimofte | K-decompositions and 3d gauge theories[END_REF]. The definition is related to the case of decorated triangulated surfaces [START_REF] Fock | Moduli spaces of convex projective structures on surfaces[END_REF]. The decoration corresponds to an assignment of a configuration of four complete flags in C n to each 3-simplex of the triangulated the 3-manifold satisfying certain compatibility conditions. The flags in the configuration can be thought as associated to each of the four vertices of a simplex. We will review mainly the case n = 3 [BFG, FKR]. Beware that the presentation given here is suitable for n = 3 and uses the coordinates described in the previous section. But it is not the best set of coordinates for higher dimensional generalizations [START_REF] Garoufalidis | Gluing equations for PGL(n, C)representations of 3-manifolds[END_REF][START_REF] Dimofte | K-decompositions and 3d gauge theories[END_REF][START_REF] Bucher | The bounded Borel class and 3-manifold groups[END_REF]. The decorated triangulations give rise to representations of the fundamental group into PGL(3, C) or SL(3, C) and the goal is to find, among these representations, those with values in one of the real forms.

6.1. Decorated triangulations of 3-manifolds and representations. We consider an ideal triangulated compact oriented 3-manifold M with boundary ∂M . It means a finite number of oriented 3-simplices T ν (1 ≤ ν ≤ N ) and an orientation reversing matching of their faces so that, if one truncates all the simplices at their vertices and identifies the truncated faces, one obtains a manifold homeomorphic to M with ∂M identified to a triangulated 2-surface.

Let V = C n and F the flag variety of complete flags. We let G = PGL(n, C) be the projective group acting on the space of flags. Definition 6.1. A decoration of an ideal triangulation of a manifold M is a collection of maps from the vertices of each tetrahedron T ν to the space of configurations F 4,gen /G compatible with the matchings. More precisely, to the 0-simplex of T ν one associates a configuration of flags up to the action of G such that two indentified faces are associated with the same point in F 3,gen /G.

We impose in the following that the decorations are chosen such that the configurations in F 3,gen have trivial stabilizer. This condition is clearly satisfied for generic configurations as defined in the previous section.

We also assume that the holonomy around the 1-skeleton be trivial. In order to understand this notion of holonomy, let us begin with a simple remark. Consider tetrahetron T , and its decoration which is an element of F 4,gen /G. Choosing a lift in F 3,gen for one face determines a lift in F 4,gen , because the stabilizer for the face is trivial. Hence, if we fix a lift for one face, we fix the lift for every face. Turning back to the holonomy around the edges: around each edge one chooses an abutted face and consider a lift in F 3,gen of the element in F 3,gen /G associated to the face. Turning around the edge, we fix a lift in F 3,gen for each abutting face. So we construct a sequence of flags corresponding to the vertices (not contained in the edge) of the faces of each of the simplices around the edge. After completing a turn around the edge we suppose that the we recover the same original configuration in F 3,gen (see Figure 4). ρ : π 1 (M ) -→ PGL(n, C). Indeed, choose a base face and consider a configuration in F 3,gen associated to it (a lift of the element in F 3,gen /G associated to the face). Any closed path γ (based at the chosen face and not intersecting any edge) determines a sequence of configurations in F 3,gen corresponding to the configurations for each successive face crossed by the path. The last face configuration of flags along the path is the image of the reference configuration of flags by a unique element in G. This element is called the holonomy along γ. Clearly, if the holonomy around all edges are trivial, the paths may cross edges without modifying the holonomy and therefore the holonomy representation is well defined on π 1 (M ).

From the previous section, we see that a generic decoration gives coordinates z ν ij for each tetrahedron. One can write algebraic equations which describe the matching between the faces and the triviality of the holonomy around the edges easily. In order to match a face between two tetrahedra T ν and T µ one observe that the three flags (indexed by i, j, k) have opposite orientation when viewed with the induced orientation on each tetrahedron. One obtains that the triple ratios, which classify triples up to the action of PGL(3, C), satisfy z ν ijk = 1/z µ ikj . On the other hand, a simple computation with cross-ratios verifies, taking account of all tetrahedra T ν having a common edge ij, the following conditions

ν z ν ij = ν z ν ji = 1.
6.2. Unipotent decorations. It is important to observe that the holonomy along any path γ on a boundary component of M corresponding to a vertex of the triangulation preserve the flag associated to this vertex. Indeed, all the faces considered during the computation of the holonomy share this vertex. Hence, the holonomy fixes the flag at this vertex. Up to conjugation the holonomy is therefore contained in a Borel group.

An important class of decorations is obtained by adding the condition that the holonomy of any boundary path is conjugate to a unipotent radical of a Borel subgroup of G. In the case of SL(n, C) or PGL(n, C), for each boundary component, the boundary holonomy may then be considered to consist of upper triangular matrices with equal entries in the diagonal.

One can compute holonomies in terms of the coordinates z ν ij . Imposing that the boundary holonomy be unipotent can be translated into algebraic equations. One obtains a system of algebraic equations, but due to the the restriction on the coordinates to be generic the solution set is semi-algebraic.

Computations of decorations with complete flags were implemented in [C] (see also [FKR]). Unipotent decorations were computed for PGL(2, C), PGL(3, C) in many cases and for one manifold, the complement of the figure eight knot, for PGL(4, C).

6.3. G 0 -decorations and representations in real forms. Definition 6.2. Let G 0 be a real form of G and F 0 the closed G 0 -orbit in F . A G 0 -decoration of an ideal triangulation of M is a decoration obtained through a family of maps from each tetrahedron to F 4,gen 0 /G 0 (here F 4,gen 0 = F 4,gen ∩ F 0 ).

The holonomy representation representation ρ : π 1 (M ) → G 0 will be G 0 -valued in this case.

In order to obtain explicit G 0 -decorations we use coordinates for generic flags, excluding certain degenerate configurations. For a given triangulation of a 3-manifold this might result in a loss of certain representations of its fundamental group but passing to a barycentric subdivision one obtains all representations. In generic coordinates, the G 0 -decorations are described as a constructible set given by algebraic equations implying face matchings and trivial holonomy around edges. The condition that the decoration be unipotent is described by additional algebraic equations. We will not reproduce these equations here but refer to references [START_REF] Bergeron | Tetrahedra of flags, volume and homology of SL(3)[END_REF][START_REF] Garoufalidis | Gluing equations for PGL(n, C)representations of 3-manifolds[END_REF][START_REF] Dimofte | K-decompositions and 3d gauge theories[END_REF].

A simple condition can be used to distinguish those representations which are in a real form. First of all, if all coordinates z ν ij are real, the representation is SL(3, R)-valued. For the other non-compact real form we have: Proposition 6.3 ( [BFG]). Let {T ν } with the data ν → F 4,gen /G be a decorated triangulation and let z ν ij be the coordinates of each decorated tetrahedron. If, for all ν and {i, j, k, l} = {1, 2, 3, 4}, we have

z ν ij z ν ji = z ν kl z ν lk ,
then the holonomy is PU(2, 1)-valued.

Unfortunately, computations are difficult and were completed only for triangulations with at most four tetrahedra in the case of PGL(3, C) or two tetrahedra in the case of PGL(4, C). The case PGL(3, C), for manifolds obtained with less than 3 simplices, was treated in [FKR] (see also [CURVE] for a census of manifolds triangulated by four tetrahedra).

6.3.1. The Bloch invariant. Denote an ideal triangulation of a compact manifold M by K = ν T ν and decoration by the coordinates z = (z ν ij ).

Definition 6.4. To a decorated triangulation (K, z) with coordinates z = (z ν ij ) one defines the element

β(K, z) = [z ν ij ] ∈ P(C).
Changing triangulations through 2 -3 or 1 -4 moves does not change the invariant in P(C) by Theorem 5.9.

It was proved in [BFG] that if the decorated triangulation has unipotent boundary holonomy then the invariant has values in the Bloch group.

In the case of a G 0 -decoration we obtain:

Proposition 6.5. If (K, z) is a G 0 -triangulation then β(K, z) ∈ P(C) + .
The proposition is clear for G 0 = SL(3, R). In order to show the proposition for G 0 = PU(2, 1) it is convenient to use a different set of coordinates. Namely, for a fixed tetrahedron with coordinates z ij define

w ij = z ij z ji .
Clearly for G 0 = PU(2, 1) one has w ij = wkl . Observe then that the triple ratio has a simple formula in those coordinates -z 21 z 31 z 41 = 1 w 34 w 41 w 13 , and similarly for the other triple ratios. One can show that in P(C) we have (see [START_REF] Falbel | Duality and invariants of representations of fundamental groups of 3-manifolds into PGL(3,C)[END_REF]): .

2β(K, z) = [
The proposition follows from this expression as, except from the boundary term, β(K, z) is in P(C) + and we assumed that the boundary terms cancel out. 6.3.2. Two Questions. Given a decorated ideal triangulation of a 3-manifold M , we can get a P(C)-valued invariant β(ρ) for the associated representation ρ : π 1 (M ) → P GL(n, C).

Question 1 Let G 0 be a real form with closed orbit. Suppose the decoration is a G 0 -decoration as in the Definition 6.2. Is it true that the P(C)-valued invariant β(ρ) is contained in the subgroup P(C) + ?

For n = 2, G 0 = P GL(2, R) and it is clearly true. For n = 3, G 0 = P GL(3, R) or P U (2, 1). In [START_REF] Falbel | Duality and invariants of representations of fundamental groups of 3-manifolds into PGL(3,C)[END_REF], we proved that for P U (2, 1) the invariant β(ρ) lies in P(C) + . Hence it is true for n = 3.

Question 2 Let G 0 be a real form with closed orbit. Which elements of B(C) + and B(R) can be realized as the invariant β(ρ) for ρ coming from a G 0 -decoration? Note by Proposition 4.8, the difference between B(C) + and B(R) is the torsion part of B(R).

For n = 3, we have some examples which realize some elements in B(R).

Example 6.6. The 5 2 knot complement. Consider the real root γ of the equation

x 3 -x + 1 = 0. By [FKR] pages 252-253, the invariant β(ρ) = 12[γ]. It is easy to check that λ(β(ρ)) = 12γ ∧ (1 -γ) = 0. So β(ρ) ∈ B(R)
. By the formula (4.2.3), we can numerically compute the value

12L( 1 1-γ ) π 2 .
We find it is irrational numerically. This suggests that β(ρ) is of infinite order.

Example 6.7. The figure-eight knot's sister manifold. By [BFGKR] page 419, there exist two 1-dimensional components of real solutions. Let τ = 1 2 ± 1 2 √ 5. Put X = τ Y . The first component is given by

z 12 = w 21 = X+Y X-1 ; z 21 = w 12 = 1 + Y ; z 34 = w 43 = X 2 +X+Y
X(X-1) ; z 43 = w 34 = X. Hence we have the invariant

β 1 (Y ) = 2([τ Y ] + [1 + Y ] + [ τ 2 Y τ Y -1 ] + [ τ (Y + 1) τ Y -1 ]). We can check that β 1 (Y ) = 8[τ 2 ] ∈ B(R) which is independent of Y .
The second component is given by

         z 12 = w 21 = 1 + Y X -(X+1)(Y +1) X 2 +X-1 z 21 = w 12 = X+Y -1 Y -1 z 34 = w 43 = X + Y z 43 = w 34 = 1 Y Hence we have the invariant β 2 (Y ) = 2([ τ 2 (Y -1) τ 2 Y -1 ] + [ τ 2 Y -1 Y -1 ] + [τ 2 Y ] + [ 1 Y ]). We can check that β 2 (Y ) = 2[τ 2 ] ∈ B(R). Let τ + = 1 2 + 1 2 √ 5 and τ -= 1 2 -1 2 √ 5
. By Example 4.5, we know that both [τ 2 + ] and [τ 2 -] have order 10 in B(R). Therefore, both β 1 (Y ) and β 2 (Y ) are non-trivial torsion elements of B(R), whose orders are equal to 5. Remark 6.8. The Bloch invariant is related to the Borel invariant: We are given a representation ρ : π 1 (M ) → PSL(3, C), of the fundamental group of a compact manifold M which we suppose is a K(π 1 (M ), 1)-space. In particular H 3 (M, R) H 3 (π 1 , R). We let β c be a generator of the continuous cohomology H 6.3.3. Geometric structures associated to a decorated representation. The computations with coordinates of configurations of flags G 0 -decorations give rise to representations into the real forms PU(2, 1) and SL(3, R). In both cases, one can, in certain cases, associate (G 0 , F 0 )-structures to the 3-manifold such that its holonomy coincides with the representation. We refer to [Fa, FS] for more details. A basic question is Question Which 3-manifolds have geometric structures modeled on a real form G 0 ?

Chern-Simons invariants

In this section we recall three invariants related to Chern-Simons invariants: one for flag structures, following the Riemannian construction, a second one for representations, via the construction of a flat G-bundle and a third one for representations via characteristic classes.

We conclude by the relations -known and less known -relations between these constructions.

7.1. The Chern-Simons invariant of a flag structure. The first invariant is related to the Chern-Simons invariant of a Riemannian manifold. It is defined for flag structures, not necessarily flat ones.

This construction follows the classical Chern-Simons construction in [CS] and, in the case of CR structures [BE] (see [FV]).

We let M be a real closed 3-manifold equipped with a flag structure. We denote Y the canonical bundle defined above. The second Chern class of the bundle Y with connection form π is given by

c 2 (Y, π) = 1 8π 2 T r(Π ∧ Π). In the case of the connection form π we obtain   0 -Φ 2 -1 4 Ψ 0 0 1 2 Φ 1 0 0 0   ∧   0 -Φ 2 -1 4 Ψ 0 0 1 2 Φ 1 0 0 0   =   0 0 -1 2 Φ 1 ∧ Φ 2 0 0 0 0 0 0   . As Φ 1 = Q 1 ω ∧ ω 2 and Φ 2 = Q 2 ω ∧ ω 1 we have Π ∧ Π = 0 and c 2 (Y, π) = 0.
One observes that the form In principle that integral depends on the section but the following proposition shows that the integrand

T C 2 (π) = 1 8π 2 T r(π ∧ Π) + 1 3 T r(π ∧ π ∧ π) = 1 24π 2 T r(π ∧ π ∧ π)
s * T C 2 (π)
defines an element in the cohomology which does not depend on the section. Proposition 7.2. Suppose s and s are two sections. Let h : M → H be such that s = R h • s. Then

s * T C 2 (π) -s * T C 2 (π) = - 1 8π 2 d s * T r(h -1 π ∧ d h).
Let µ(t) be the invariant as a function of the a parameter describing the deformation of the structure on a closed manifold M and define δµ = d dt µ(0). The following proposition shows that the set of critical points of this invariant contains the flag structures with zero curvature: Proposition 7.3. δµ = -1 4π 2 M s * T r( π ∧ Π). In particular, µ is constant on real flag structures associated to a homotopy class of totally real immersions on the complex flag space.

7.2. The Chern-Simons invariant of a representation. When dealing with a geometric structure on a manifold one obtains a holonomy representation. But, even when the structure is not defined and one is given a representation ρ : π 1 (M ) → G one can construct an invariant through the flat G-bundle associated to ρ. It is defined to be The invariant above does not depend on the section once we verify that the integration of the form CS(π) along the fiber is an integer.

E ρ = M × ρ G = M × G/(x, g) ∼ (hx, ρ(h -1 )g).
Remark 7.4. In the presence of cusps one can also define an invariant ([M]).

7.3. The Cheeger-Chern-Simons invariant of a representation. Cheeger-Chern-Simons classes arises from characteristic classes of universal bundles. In the case of PSL(2, C) (mostly related to hyperbolic geometry) one obtains the map from the third homology of the group considered as a discrete group (see [START_REF] Cheeger | Differential characters and geometric invariants[END_REF]):

ĉ2 : H 3 (PSL(2, C), Z) → C/π 2 Z.

If ρ : π 1 (M ) → PSL(2, C) is a representation of the fundamental group of a K(π 1 (M ), 1)-space we have H 3 (M, Z) = H 3 (π 1 (M ), Z) and therefore the fundamental class [M ] can be seen inside H 3 (π 1 (M ), Z) and its image through H 3 (π 1 (M ), Z) → H 3 (PSL(2, C), Z) composed with ĉ2 gives rise to an invariant iV ol(ρ) + 2π 2 cs(ρ)), where V ol(ρ) ∈ R and cs(ρ) ∈ R/Z. We define V ol C (ρ) = -iĉ 2 to be the complex volume of ρ.

In the case of the simply connected group SL(2, C) one has An explicit description of the map ĉ2 is based on the dilogarithm as a function on the Bloch group (See [D, N, BFG, GTZ]).

Equalities between invariants.

As for the relation between the last two Chern-Simmons invariants for representations, one can prove that cs(ρ) coincides with the Chern-Simons invariant defined above in the case the associated PSL(2, C)bundle is trivial. Observe, though, that as 2π 2 cs(ρ) ∈ C/Z the complex volume misses some information carried by cs (M ).

The relation to the first invariant is less known. In general, given a representation, one does not know whether it is the holonomy of a flat structure. However, if this is the case, one can obtain a relation between the invariants. The proof of the following proposition is as in ( [Kh]) for the case of a spherical CR structure. The real volume can be computed from the coordinates z ij through the Bloch invariant of the decoration, as the sum of the Bloch-Wigner dilogarithm [BBI]. A variant of it, involving flattenings, allows to compute the complex volume and hence the Chern-Simmons invariant.

If the representation takes values in a real form G 0 , then one can prove that the real volume is 0 [BFG] but it is not clear what happens to the Chern-Simmons invariant in this case.

Through the database curve [CURVE], one can look at values of the complex volume of unipotent decoration. An observation that can be done is the following one. Each unipotent decoration takes values in a number field, hence gives rise to a family of Galois conjugates. One can check experimentally that, if all the Galois conjugates are inside one real form then the Chern-Simmons invariant cs(ρ)

3. 1 .

 1 Real submanifolds in flag space. Let φ : M → SL(3, C)/B be an embedding and φ * : T M → T SL(3, C)/B be its differential. Extend this map to the complexification T M C of T M defining φ * : T M C → T SL(3, C)/B by φ * (iv) = Jφ * (v), where J is the complex structure on the tangent space of the complex manifold SL(3, C)/B. Definition 3.1. An embedding φ : M → SL(3, C)/B is totally real if, for every p ∈ M , φ * : T p M C → T p SL(3, C)/B is an isomorphism.

Figure 1 .Figure 2 .

 12 Figure 1. The cross ratio of four lines passing through a point p

  Proposition 5.7. A generic tetrahedron of flags T ∈ Gc 4 is uniquely determined by the 4 edge coordinates (z 12 (T ), z 21 (T ), z 34 (T ), z 43 (T )) which are in C \ {0, 1}. Hence Gc 4 (C \ {0, 1}) 4 .

Figure 3 .

 3 Figure 3. The z-coordinates.

Figure 4 .

 4 Figure 4. tetrahedra sharing a common edge

  3 c (PSL(3, C), R) R and ρ * : H 3 c (PSL(3, C), R) → H 3 (π 1 , R) be the pull-back of cohomology classes. We define then B(ρ) = ρ * (β c )([M ]), where [M ] is the fundamental class of M . When M has cusps H 3 (M, R) = 0 and, in [BBI], the authors were able to represent β c by a bounded cocycle defining an element β b ∈ H 3 cb (PSL(3, C), R) where H * cb refers to bounded continuous cohomology. They prove that, for a hyperbolic manifold with cusps, B(ρ) is maximal if and only if ρ is the geometric representation obtained by composition of the geometric representation of the fundamental group into SL(2, C) with the irreducible representation of SL(2, C) into PSL(n, C).

s

  is closed and 0 = c 2 (Y, π) = d T C 2 (π).Definition 7.1. Suppose that the fiber bundle Y → M is trivial and let s : M → Y be a section, we define then * T r(π ∧ π ∧ π).

  Let π : T E ρ → g be the flat connection on E ρ associated to the representation. π ∧ π ∧ π) .Suppose now that E ρ (that is the case if G = SL(n, C)) is trivial and define for each section of the bundle cs(ρ) = M s * CS(π) mod Z.

  ĉ2 : H 3 (SL(2, C), Z) → C/4π 2 Z.If ρ : π 1 (M ) → SL(2, C) is a lift of a geometric representation of a closed hyperbolic manifold one obtains that ĉ2 (ρ) = i(V ol(M )+i2π 2 cs(M )) as an element in C/2π 2 Z (theorem 2.8 GTZ). Here V ol(M ) is the hyperbolic volume of M .Similarly one can define ĉ2 : H 3 (PSL(n, C), Z) → C/π 2 Z and ĉ2 : H 3 (SL(n, C), Z) → C/4π 2 Z. and the corresponding complex volume invariants of representations ρ : π 1 (M ) → PSL(n, C) and ρ : π 1 (M ) → SL(n, C) respectively.

  Proposition 7.5. Let M be a 3-manifold with a flat flag structure with canonical bundle Y → M which we suppose trivial. Denote the holonomy map of the flat structure ρ : π 1 (M ) → G. Then µ(M ) = -cs(ρ) mod Z. 7.5. G 0 -representations and Chern-Simmons. Consider a representation of the fundamental group of a triangulated manifold M to PGL(3, C), associated to a decoration.

  w 12 ]+[w 21 ]+[w 34 ]+[w 43 ]-([w 12 w 13 w 23 ]+[w 13 w 34 w 41 ]+[w 14 w 42 w 21 ]+[w 24 w 43 w 32 ])+R

	where R is the sum of the following terms
	[	w 12 w 14 -w 12 + 1 w 12 w 14	] + [	1 w 24 w 12 -w 24 + 1	]
	[	w 12 w 23 -w 12 + 1 w 23 w 12	] + [	1 w 31 w 12 -w 31 + 1	]
	[	w 34 w 23 -w 34 + 1 w 23 w 34	] + [	1 w 24 w 34 -w 24 + 1	]
	[	w 34 w 14 -w 34 + 1 w 34 w 14	] + [	1 w 13 w 34 -w 13 + 1	]

5.1. Cross ratio. Let F be a field. Let V be a vector space over F . Let P(V ) be the projective space of V , that is, P(V ) is the set of 1-dimensional subspaces of V .

Suppose dim(V ) = 2 and we fix a basis of V . Given four distinct ordered elements l i = [a i , b i ] ∈ P(V ), 1 ≤ i ≤ 4, where [a i , b i ] are the corresponding homogeneous coordinates.

Definition 5.1. We define the cross ratio of the four distinct ordered elements l i ∈ P(V ) by the formula:

where |l i , l j | denotes the determinant of the 2 × 2 matrix a i a j b i b j .

It is well-known that the cross ratio is independent of the choices of the basis of V and it is invariant under the group PGL(V ). In particular, if

Moreover, it is straightforward to check the following properties of the cross ratio, and we will omit the proof.

Lemma 5.2. (1) Let c = X(l 1 , l 2 , l 3 , l 4 ), then we have the following identities:

(2) Let K 4 = {1, (12)(34), (13)( 24), (14)(23)} which is a normal subgroup of S 4 . Then ∀σ ∈ K 4 , X(l σ(1) , l σ(2) , l σ(3) , l σ(4) ) = X(l 1 , l 2 , l 3 , l 4 ).

Now suppose dim V = 3 and consider the projective plane P(V ). For a point p ∈ P(V ), we define L(p) to be the set of all the (projective) lines in P(V ) passing through p. To avoid any confusion, we will denote by l p the 1-dimensional subspace of V represented by p. Clearly L(p) can be naturally identified with the set of all 2-dimensional subspaces of V which contain l p . Let V * be the dual space of V . Let

Moreover, L(p) can be naturally identified with P(V * (p)). Hence it is a projective line. Given 4 distinct ordered lines l i ∈ L(p), 1 ≤ i ≤ 4, we will denote their cross ratio by X(p|l 1 , l 2 , l 3 , l 4 ).

The following proposition is well-known (see Figure 1).

where the cross ratio on the right-hand side is on the line L.

The following proposition will be needed in the proof of the main Theorem 5.9 in this section. It relates the cross ratios of four lines passing through a point in a conic and the cross ratio of four other lines passing through a different point but intersecting the conic at the same points. See Figure 2 for the illustration. It is equivalent to Pascal's theorem. is a simple rationnal number -or equivalently, the complex volume is a rationnal multiple of 2iπ 2 . This question raises several interesting questions largely unstudied at this point, related to Galois actions on B(C).