Fadhila Chehami
email: fadhila.chehami@ummto.dz

Zahia Kebci

Abderrahmane Belkhir

• Fadi

Issam Baida

Omar Lamrous

F Baida

Numerical study of an efficient light focusing nano-coupler based on C-shaped waveguides

Keywords: C-shaped waveguide, Nano-coupler, Light focusing, FDTD simulation, Coupling efficiency

We present an FDTD simulation study of a new design of light focusing nano-coupler consisting of a tapered succession of C-shaped silver waveguides. Its operation is based on the use of the fundamental guided mode T E 10 . Thanks to the optimization of the structural design parameters, we have demonstrated numerically that this compact coupler enables a sub-wavelength light focusing with a transmission coefficient of 65% and a coupling efficiency of 77% that have

Introduction

Photonic integrated circuits have greatly progressed in the last decade replacing the electronic ones, due to their unique ability to transport huge amounts of information over long distances. In this respect, they were developed in order to connect different optical devices achieving great improvements in rapidity, compactness and profitability [1,2]. Studies show, however, that during the light transmission from an optical fiber at micrometer level to an integrated waveguide at nanometer level, very high losses occur because of the guiding properties mismatch existing between the two devices. Therefore, it turned out necessary to develop devices called couplers, which play the role of optical funnels. These are destined to concentrate the light coming from a large to a small area with minimum optical losses thus maximizing the performances of photonic devices such as detectors [3,4], modulators [5,6] and light emitting diodes [7,8]. That is why there is a number of studies related to this topic [9,10]. Depending on the employed physical approach, the developed nano-coupler devices could be divided in four major classes : direct couplers, tapered waveguides, lens couplers and scatterers.

The simplest way is the direct coupling, in which the light is coupled first to a wide waveguide and then, with the help of a directional coupler, coupled into a smaller nano-waveguide. This kind of coupler which is based on end fire coupling [11,12], resonant stub [13,14] or arranging similar waveguides one next to or inside another [15][16][17], allows to achieve a large coupling efficiencies of about 70% but, since it requires preliminary coupler, the total coupling efficiency is reduced to 40%. The coupling phenomenon was also achieved using the tapered waveguides in which the light is compressed when propagating through the taper. The proposed optical funnel containing a metal-dielectric photonic crystal in [18] reaches only 13% of transmission and the tapered metallic wire proposed in [19] and experimentally fabricated in [20] shows a transmission of 20%. In addition, light could be tightly focused using dielectric lenses that achieve a maximal coupling efficiency of 45% [21] or plasmonic ones leading to a low transmission not exceeding 30% [22]. In order to achieve an effective coupling, other studies used scatterer couplers based on antennas or diffraction gratings. On the one hand, the antenna couplers exhibit a theoretical limit of 50% of the coupling efficiency due to the reradiation of the captured power back into free space [START_REF] Balanis | Antenna theory: analysis and design[END_REF]. Indeed, in the work [START_REF] Huang | [END_REF], only a 10% coupling efficiency is reached at λ = 1550 nm. On the other hand, the diffraction gratings exhibit high coupling efficiency up to 70% to the detriment of compactness because of their relatively large size [25,26].

In this work, we propose a new design of a light focusing nano-coupler consisting of a tapered succession of silver C-Shaped WaveGuides (CSWG s ). We start, in Section 2, by studying the influence of the different geometrical parameters of a CSWG on the cutoff wavelength of its T E 10 fundamental guided mode. In Sec-tion 3, we present the simulation results of the optimized coupler in the case of a periodic grating and as an isolated pattern.

2 Study of the TE 10 fundamental mode of a CSWG The scheme of a CSWG is shown in Fig. 1. It consists of a coaxial waveguide made of silver metal, having an outer radius R o and an inner radius R i with an additional metallic part controlled by the angle β. This structure was used for the conception of our couplers since the breaking of the revolution symmetry compared to the case of conventional coaxial waveguides enables the red shift of the cutoff wavelength allowing the realization of compact metamaterials exhibiting original properties such as negative refraction index [27] or artificial anisotropy [28]. Moreover, compared to other waveguide geometries, the C-Shaped geometry allows the propagation of the optical signals over quite long distances of the order of several tens of micrometers without much attenuation via its T E 10 fundamental mode [29]. In this study, the numerical calculations were performed using the Finite Difference Time Domain (FDTD) algorithm previously developed [30]. FDTD is a numerical method to solve Maxwell's equations, using spatial and temporal discretizations that allow determining the electromagnetic field [31]. Note that our homemade FDTD code was improved by integrating the Drude model in order to take better account of the metal dispersion [32][33][34]. Here, silver is preferred over other metals in order to reduce the propagation losses in the optical range. The structure eigen-modes and the corresponding eigen-values are then deduced from the resolution of the electromagnetic wave equation. First, as shown in [29], we confirmed that the fundamental guided mode of the CSWG is the T E 10 mode. This latter is similar to the fundamental mode of a rectangular-section waveguide and it is characterized by a lobe of E-field at its center. It is then perpendicularly polarized to the waveguide long sides (curved sides), therefore parallel to the axis of symmetry of the section. To do this, we calculated the spectral energy density of the structure [35,36] depicted in Fig. 2, when R i = 140 nm, R o = 240 nm and β = 177 • . For these geometrical parameters, we get a cutoff wavelength of λ c = 1550 nm. The inset in Fig. 2 depicts the electric intensity distribution of the T E 10 mode at the cutoff. As it can be seen, the latter displays only one lobe of electric intensity which confirms the excitation of the T E 10 fundamental mode. It should be noted that, for the same radii, the CSWG handles guided mode with large value of wavelength compared to the conventional coaxial one. This opens the way for new perspectives in the device miniaturization and for enhanced transmission applications [29]. After that, we studied the influence of the geometrical parameters (R i , R o and β) on the cutoff wavelength of the T E 10 fundamental mode. First, we vary the outer radius R o of a CSWG with fixed inner radius and -Non linear variation, when R o is close to R i , due to the coupling between the two plasmonic metallic interface modes. This leads to an effective mode which penetrates more inside the metallic parts of the waveguide thus increasing the metallic losses.

-Linear variation, when R o is far from R i , corresponding to the case in which the coupling between the two plasmonic metallic interfaces modes no longer happens.

Moreover, this behavior was already observed in the case of coaxial waveguides [37]. Then, the cutoff wavelength of the T E 10 mode was calculated as a function of β when R i = 140 nm and R o = 240 nm. As illustrated in Fig. 3b, the cutoff wavelength decreases almost linearly with the increase of the additional metallic part size. Therefore, the cutoff wavelength of the T E 10 mode could be adjusted through the different geometrical parameters of the CSWG. This enables to control the cutoff of the different CSWG s that we will use to design our coupler device.

Design of the light focusing coupler

Our sub-wavelength light focusing coupler, shown in Fig. 4, consists of a tapered succession of four CSWG s on which a gradual decrease on their radii is imposed leading to the following radii : CSWG

1 (R i = 140 nm, R o = 240 nm), CSWG 2 (R i = 120 nm, R o = 220 nm), CSWG 3 (R i = 100 nm, R o = 200 nm) and CSWG 4 (R i = 80 nm, R o = 180 nm).
It should be noted that the different radii were picked in order to arrange the different metallic apertures one in front of another thus minimizing the reflections near the connection areas. Note that all the numerical calculations that will follow were performed by illuminating the coupler with a plane wave linearly polarized parallel to the CSWG's axis-symmetry (here the x axis) and propagating along the z axis (see Fig. 4).

At the outset, in order to get all these CSWG s exhibiting the same cutoff wavelength at λ = 1550nm, we proceeded to the optimization step using the following analytical formula [29] :

λ c = 2(π -β/2)(R o + R i ) + α.
This latter is only valid in the linear variation zone observed in Fig. 3a corresponding to outer radii far from the inner ones which is the case in our structures. α is the red-shift of the T E 10 fundamental mode cutoff due to the fact that our CSWG s are made of silver at which the light could penetrate unlike the case of a perfect electric conductor (PEC) at which the light is entirely reflected. The value of α = 330 nm is deduced from the previous formula using the geometrical parameters of the CSWG studied in Fig. 2

Study of the light focusing coupler as a periodic grating

First, we considered our coupler as a periodic grating of the pattern shown in Fig. 4 having a period p = 700 nm along the x and y directions. This periodic arrangement enables the transmission enhancement since this latter is an individual effect (guided mode). In fact, the period of the grating is chosen so that the cutoff wavelength of the T E 10 fundamental mode is larger than the Rayleigh anomaly wavelength [38] given by : λ R = n × p where n represents the substrate refractive index (in our case, n = 1). In order to pick up the optimal coupler's height ensuring the most efficient coupling, we fixed all the geometrical parameters of the grating coupler and we calculated the transmission coefficient (T ) and the coupling efficiency (CE) corresponding to several coupler's heights at λ = 1550 nm. The coupling efficiency is calculated by dividing the power flow calculated through the CSWG 4 section constituting the top of the coupler by the one calculated through the CSWG 1 section constituting the base of the coupler. Figs. 5a andb display the variation of the transmission coefficient (T ) and the coupling efficiency (CE) respectively as a function of the grating coupler's height (h c ). When looking to the latter, one can see that, from h c = 500 nm to h c = 1500 nm, the proposed grating coupler exhibits a good coupling performances reaching T = 65% and CE = 77%. These original coupling properties come from the spatial compression of the T E 10 fundamental mode when propagating through the tapered succession of the CSWG s thanks to the appropriate proposed juxtaposition ensuring minimal losses. However, we can clearly notice that increasing the coupler's height beyond h c = 1500 nm lowers the transmission coefficient and the coupling efficiency under 50% since the metallic losses increase when increasing the propagation length. In Fig. 6, we show the spectral variation of the power flows calculated considering a grating coupler with p = 700 nm and h c = 1000 nm. P 1 is the power flow calculated through the CSWG 1 section constituting the base of the coupler and P 4 is the one calculated through the CSWG 4 section constituting the top of the coupler. These latters exhibit three different peaks at λ = 1550 nm, λ = 1364 nm and λ = 1050 nm. So, in order to explain the origin of these three peaks, we mapped, in Fig. 7, the electric intensity distributions in the horizontal and vertical plans corresponding to each peak. On the one hand, we notice that the three xy electric intensity distributions (Figs. 7a,b andc) exhibit one lobe of electric intensity corresponding to the excitation of the T E 10 fundamental mode. On the other hand, the xz and the yz electric intensity distributions calculated at λ = 1550 nm show an uniform illumination due to the constant phase during the back and forth of the light propagation (See Figs. 7d andg). However, the xz and yz electric intensity distributions calculated at λ = 1364 nm and λ = 1050 nm depicted in Figs. 7(e,h) and (f, i), exhibit one and two intensity nodes corresponding to the first and the second harmonic of Fabry Perot resonances of the same T E 10 fundamental mode respectively. Therefore, we are able to consider our proposed structure as an efficient coupler operating at various wavelength values which can be controlled by modifying the different geometrical parameters of the CSWG s . In addition,

Study of the light focusing coupler as an isolated pattern

In this part, to validate the efficiency of our coupler as an isolated structure, we consider only one coupler pattern (shown in Fig. 4) engraved into an infinite silver layer. Then, we calculated the variation of the coupling efficiency (CE) and the transmission cross section (T c ) as a function of the isolated coupler height (h c ) (See Fig. 8). The calculation of CE has been carried out following the same steps as in the periodic case. However, the transmission cross section has been calculated by dividing the power flow reaching the far field from the end of the coupler (in the z direction) by the one penetrating into the CSW G 1 section constituting the base of the coupler. It should be noted that instead of the transmission coefficient, the transmission cross section has been calculated in this part because, in the case of an isolated pattern, the transmission quantification is carried out, as is cus- tomary, by calculating its cross section. Moreover, considering the isolated coupler having (h c = 1000nm), we showed in Fig. 9 the power flows P 1 and P 4 (calculated through the CSWG 1 and CSWG 4 constituting the base and the top of the isolated coupler respectively) and we mapped in Fig. 10 the electric intensity distributions in horizontal and vertical plans at λ = 1553 nm. When comparing Fig. 5 with Fig. 8 and Fig. 6 with Fig. 9 and Figs. 7a, d and g with Figs. 10a, b andc, we can clearly notice that both the periodic grating coupler and the isolated one exhibit similar coupling behavior. All these results demonstrate that the proposed coupler operates at various wavelengths in both arrangements. The slight differences that might be noticed are due to the fact that in the periodic grating, we observe the coupling effect of several holes unlike the isolated pattern in which we see the effect of only one hole engraved in an infinite metallic layer [39,40]. Moreover, as it is highlighted in Table 1, our proposed structure exhibits important coupling properties compared to the other coupling approaches previously reported in other studies.

Conclusion

In summary, we studied the excitation and the propagation of the T E 10 fundamental mode of a CSWG and showed, as expected, that its cutoff is highly sensitive to the different geometrical parameters. Then, we exploited the latter for designing an original optical coupler exhibiting a high sub-wavelength light focusing, at telecom wavelengths, with good transmission coefficient (T = 65%) and coupling efficiency (CE = 77%). Moreover, we demonstrated that in addition to being a functional structure, operating at several wavelengths, our proposed coupler shows

Fig. 1

 1 Fig. 1 Scheme of a silver CSWG having an inner radius R i , an outer radius Ro and a additional metallic part β.

Fig. 2

 2 Fig. 2 Normalized energy density of a CSWG with R i = 140 nm, Ro = 240 nm and β = 177 • . The inset shows the calculated electric intensity distribution (fifth root of the square modulus of the electric field) at the cutoff λc = 1550 nm. The dashed lines underline the CSWG section.

Fig. 3

 3 Fig. 3 Variation of the T E 10 cutoff wavelength as a function of : (a) The outer radius Ro of a silver CSWG with R i = 140 nm and β = 177 • . (b) The additional metallic part β of a silver CSWG with R i = 140 nm and Ro = 240 nm.

  (with R i = 140 nm, R o = 240 nm and β = 177 • exhibiting a cutoff at λ c = 1550nm). After calculating the red-shift α and knowing the values of the different CSWG s radii, we calculate the size of the additional metallic part β specific to each CSWG constituting the coupler. This optimization leads to the following geometrical parameters: CSWG 1 (R i = 140 nm, R o = 240 nm and β = 177 • ), CSWG 2 (R i = 120 nm, R o = 220 nm and β = 155 • ), CSWG 3 (R i = 100 nm, R o = 200 nm and β = 127 • ) and CSWG 4 (R i = 80 nm, R o = 180 nm and β = 90.2 • ). We confirmed the accuracy of these results, numerically, by calculating the cutoff corresponding to each CSWG taken separately which leads to equal cutoffs at λ c = 1550nm.

Fig. 4

 4 Fig. 4 Scheme of the proposed light focusing coupler having a height hc = 1000 nm constituted by a tapered succession of four equal height CSWGs.

Fig. 5

 5 Fig. 5 (a) Transmission coefficient (T ) and (b) Coupling efficiency (CE) calculated as a function of the grating coupler's height (hc) having a period p = 700 nm at λ = 1550 nm.

Fig. 7

 7 Fig.7shows that the maximum electric intensity grows when propagating along the z axis which further emphasizes the efficiency of our coupler since it allows a quasi-conservation of the guided energy in each CSWG constituting the coupler.

Fig. 6

 6 Fig. 6 Spectral variation of the power flows calculated through the CSWG 1 section constituting the base (P 1 ) and through the CSWG 4 section constituting the top (P 4 ) of a grating coupler having p = 700 nm and hc = 1000nm. The power flows depicted were normalized by the maximal value of P 1 .

Fadhila

  Chehami et al.

Fig. 7

 7 Fig. 7 Electric intensity distributions (third root of the square modulus of the electric field) calculated in the horizontal plan xy (a, b and c) and the vertical plans xz (d, e and f) and yz (g, h and i) passing through the middle of the structure. The wavelength value is set to : λ = 1550 nm in (a,d and g), λ = 1364 nm in (b, e and h) and λ = 1050 nm in (c,f and i). The grating coupler has the following geometrical parameters: p = 700 nm and hc = 1000 nm.

Fig. 8

 8 Fig. 8 (a) Transmission cross section (Tc) and (b) Coupling efficiency (CE) calculated as a function of the isolated coupler's height (hc) at λ = 1553 nm.

Fig. 9

 9 Fig. 9 Spectral variation of the power flows calculated through the CSWG 1 section constituting the base (P 1 ) and through the CSWG 4 section constituting the top (P 4 ) of an isolated coupler having hc = 1000 nm. The power flows depicted were normalized by the maximal value of P 1 .

Fig. 10

 10 Fig. 10 Electric intensity distribution (third root of the square modulus of the electric field) of the isolated coupler having hc = 1000 nm calculated at λ = 1553 nm in the : (a) xy plan, (b) xz plan and (c) yz plan passing through the middle of the structure.

Fadhila Chehami et al. efficient performances when arranged in a periodic grating as well as an isolated pattern. The future manufacturing of this type of structure will allow us to carry out a follow-up study to develop a novel generation of metamaterials for nano guiding, optical trapping and sensing applications.

Conflict of interest

The authors declare that they have no conflict of interest.


Jean-Yves Welschinger

Spectral sequences of a Morse shelling

Keywords: spectral sequence, simplicial complex, discrete Morse theory, shellable complex, tilings. Mathematics subject classification 2020: 55T99, 57Q70, 55U10, 52C22

We recently introduced a notion of tilings of geometric realizations of finite relative simplicial complexes and related those tilings to the discrete Morse theory of R. Forman, especially when they have the property of being shellable, a property shared by the classical shellable complexes. We now observe that every such tiling supports a quiver which is acyclic precisely when the tiling is shellable and then, that every shelling induces two spectral sequences which converge to the relative (co)homology of the complex. Their first pages are free modules over the critical tiles of the tiling.

Introduction

We recently introduced [START_REF] Salepci | Tilings, packings and expected Betti numbers in simplicial complexes[END_REF][START_REF] Salepci | Morse shellings and compatible discrete Morse functions[END_REF] a notion of tiling of the geometric realization of a finite simplicial complex. It is a partition by tiles, where a tile is a maximal simplex deprived of several of its codimension one faces together with possibly a unique face of higher codimension, see §2. When the tiles can be totally ordered in a way defining a filtration by subcomplexes, the tiling is said to be shellable, or rather Morse shellable, since this notion extends the classical notion of shellability [START_REF] Bruggesser | Shellable decompositions of cells and spheres[END_REF][START_REF] Adiprasito | Subdivisions, shellability, and collapsibility of products[END_REF][START_REF] Stanley | Combinatorics and commutative algebra[END_REF][START_REF] Ziegler | Lectures on polytopes[END_REF][START_REF] Hachimori | Decompositons of simplicial balls and spheres with knots consisting of few edges[END_REF][START_REF] Benedetti | On locally constructible spheres and balls[END_REF] and is related to discrete Morse theory [START_REF] Forman | Morse theory for cell complexes[END_REF][START_REF] Salepci | Morse shellings and compatible discrete Morse functions[END_REF][START_REF] Kozlov | Organized collapse: an introduction to discrete Morse theory[END_REF]. In particular, the boundary of any convex simplicial polytope is shellable by [START_REF] Bruggesser | Shellable decompositions of cells and spheres[END_REF], but every closed triangulated surface is Morse shellable as well [START_REF] Salepci | Morse shellings and compatible discrete Morse functions[END_REF] and in fact every finite simplicial complex becomes Morse shellable after finitely many stellar subdivisions at maximal simplices or after a single barycentric subdivision by [START_REF] Welschinger | Shellable tilings on relative simplicial complexes and their hvectors[END_REF]. Not all tilings are shellable though, for any product of a sphere with a torus of positive dimension carries Morse tiled triangulations which cannot be shelled, see [START_REF] Welschinger | Morse shellings on products[END_REF].

We now observe that every Morse tiling supports a quiver which encodes its shellability, see §3.2.

Theorem 1.1. A Morse tiling is shellable if and only if its quiver does not contain any oriented cycle.

We then observe that every such shelling recovers the homology and cohomology of the complex, via two spectral sequences, see §4.

Theorem 1.2. Any Morse shelling on a finite relative simplicial complex induces two spectral sequences which converge to its relative homology and cohomology respectively and whose first pages are free graded modules over the critical tiles.

A critical Morse tile is a simplex which has been deprived of several codimension one faces together with its remaining face of maximal codimension, see §2. This terminology originates from its relation with the discrete Morse theory of Robin Forman [START_REF] Forman | Morse theory for cell complexes[END_REF][START_REF] Kozlov | Organized collapse: an introduction to discrete Morse theory[END_REF], for any Morse shelling encodes a class of compatible discrete Morse functions whose critical points are in one-to-one correspondance with the critical tiles of the shelling, preserving the index, see [START_REF] Salepci | Morse shellings and compatible discrete Morse functions[END_REF] and Theorem 5.2. As in the case of discrete Morse complexes, the chain and cochain complexes appearing in the spectral sequences given by Theorem 1.2 have much lower ranks than the simplicial ones, see Remark 4.4.

Finally, the quiver of a Morse tiled relative simplicial complex S = K \ L makes it possible to weaken the shellabiity condition into some partial shellability which implies the Morse shellability of its q-skeleton, see Theorem 3.15, and provides spectral sequences which converge to the relative (co)homology of the pair (K, L) in degrees less than q, see Theorem 4.1.

We compute in §2 the relative (co)homology of Morse tiles, showing that it does not vanish only for critical tiles. We introduce the quivers of Morse tilings in §3 and prove Theorem 1.1. We also define the notion of partial shellability and prove Theorem 3.15. We finally prove Theorem 1.2 in §4 and relate the filtration of a Morse shelling with discrete Morse theory in §5.

Acknowledgement: This work was partially supported by the ANR project MI-CROLOCAL (ANR-15CE40-0007-01).

Relative homology of a Morse tile

Let n be a non-negative integer. We recall that an n-simplex is the convex hull of n + 1 points affinely independent in some real affine space or rather, abstractly, just a set of cardinality n + 1 whose elements are vertices, see [START_REF] Munkres | Elements of algebraic topology[END_REF][START_REF] Kozlov | Combinatorial algebraic topology[END_REF]. A face of a simplex is the convex hull of a subset of its vertices and its dimension is the dimension of the affine space it spans. An open simplex (resp. face) is the relative interior of a simplex (resp. face) in its supporting affine space. The following definition has been given in [START_REF] Salepci | Morse shellings and compatible discrete Morse functions[END_REF].

Definition 2.1 (Definition 2.4 of [START_REF] Salepci | Morse shellings and compatible discrete Morse functions[END_REF][START_REF] Welschinger | Morse shellings on products[END_REF]). A Morse tile T of dimension n and order k ∈ {0, . . . , n+1} is an n-simplex T deprived of k of its codimension one faces together with one possibly empty face µ of higher codimension. It is critical of index k iff dim µ = k -1. The simplex T is called the underlying simplex while µ is called its Morse face.

A tile with empty Morse face is said to be basic and a basic tile of order k > 0 contains a unique open face of dimension k-1, its restriction set [START_REF] Stanley | Combinatorics and commutative algebra[END_REF][START_REF] Ziegler | Lectures on polytopes[END_REF], and none of lower dimension, see Figure 1. Critical tiles are thus the ones for which this peculiar face has been removed, see [START_REF] Salepci | Morse shellings and compatible discrete Morse functions[END_REF][START_REF] Welschinger | Shellable tilings on relative simplicial complexes and their hvectors[END_REF]. Note that critical tiles of maximal index are open simplices. The tiles which are not critical are said to be regular. Example 2.2. A tetrahedron deprived of one triangle and one edge is a regular Morse tile of order one and dimension three, whereas a tetrahedron deprived of two triangles and one edge is critical of index two.

Recall that a finite simplicial complex is a finite collection of simplices which contains all faces of its elements and such that the intersection of any two of them is a common face, possibly empty. Such a simplicial complex K inherits a topology whose homology and cohomology can be computed out of the simplicial chain and cochain complexes C * (K; Z) = ⊕ σ∈K Z σ and C * (K; Z) = ⊕ σ∈K hom(Z σ , Z). These complexes are graded by the dimension of the simplices and for every σ ∈ K, Z σ denotes the infinite cyclic group whose generators are the two orientations on σ, that is the kernel of the augmentation morphism aσ + bσ ∈ Zσ ⊕ Zσ → a + b ∈ Z, see [START_REF] Munkres | Elements of algebraic topology[END_REF].

Definition 2.3. The relative homology (resp. cohomology) of a Morse tile T with coefficients in the Abelian group G is the homology

H * (T , T \ T ; G) (resp. H * (T , T \ T ; G)) of the chain complex C * (T ; Z) ⊗ G/C * (T \ T ; Z) ⊗ G (resp. ker hom(C * (T ; Z), G) → hom(C * (T \ T ; Z), G) ).
Proposition 2.4. The relative homology and cohomology of regular Morse tiles vanish whereas they are isomorphic to the coefficient group and concentrated in degree k in the case of critical tiles of index k.

Proof. Let T be a Morse tile of dimension n and order k, with underlying simplex T and Morse face µ. Let σ be the union of the codimension one faces that have been removed from T , so that T = T \ (σ ∪ µ). Let θ be the convex hull of the k vertices opposite to those faces, so that θ ⊂ µ. The tile T is then regular iff θ = µ.

If T is regular, then µ contains a vertex v that does not belong to θ, so that σ ∪ µ is a cone with apex v. The inclusions v → σ ∪ µ i → T then induce isomorphisms in homology and cohomology, these being concentrated in degree zero, see [START_REF] Munkres | Elements of algebraic topology[END_REF]. The short exact sequences

0 → C * (σ ∪ µ; G) i * → C * (T ; G) → C * (T , σ ∪ µ; G) → 0 (1) and 0 → C * (T , σ ∪ µ; G) → C * (T ; G) i * → C * (σ ∪ µ; G) → 0 (2)
induce long exact sequences in homology with coefficients in the Abelian group G, from which we deduce that H * (T , σ ∪ µ; G) and H * (T , σ ∪ µ; G) vanish.

If T is critical, the result is well known for k ∈ {0, n}. When k = 0, the relative (co)homology of T is the (co)homology of a simplex while when k = n, it is the (co)homology of a simplex relative to its boundary. Otherwise, we deduce by excision that the relative (co)homology of the pair (σ ∪ µ, σ) is isomorphic to the (co)homology of µ relative to its boundary, so that it is likewise isomorphic to the coefficient group G and concentrated in degree k -1. From the long exact sequence of this pair (σ ∪ µ, σ), we deduce that the reduced homology and cohomology groups H * (σ ∪ µ; G) and H * (σ ∪ µ; G) get isomorphic to H * (σ ∪ µ, σ; G) and H * (σ ∪ µ, σ; G), so that they are isomorphic to G and concentrated in degree k -1 as well. The result then again follows from (1) and ( 2), for the reduced homology and cohomology of T vanish.

3 Quiver and shellability of a tiling 3.1 Acyclic quivers Definition 3.1. A quiver is a quadruple Q = (Q V , Q A , s, t), where Q V , Q A are finite sets whose elements are vertices and arrows, and where s, t : Q A → Q V are maps called source and target.

A quiver is thus a finite directed graph which may contain several arrows between two vertices or even a loop arrow from a vertex to itself. Example 3.2. Let (V, ≤) be a finite partially ordered set. It supports a quiver, its comparability graph, having V as set of vertices and containing an arrow for each pair (i, j) ∈ V 2 such that i ≤ j, where j is the source and i the target.

Definition 3.3. Let Q = (Q V , Q A , s, t) be a quiver. A path of Q is a p-tuple of arrows π = (a 1 , . . . , a p ) ∈ Q p A such that for every j ∈ {1, . . . , p -1}, t(a j ) = s(a j+1 ). Its source (resp. target) s(π) (resp. t(π)) is s(a 1 ) (resp. t(a p )) and its length is p.
Every vertex of a quiver also defines a path of length zero. An oriented cycle is then a path of positive length with same source and target. A quiver is said to be acyclic iff it contains no oriented cycle. Lemma 3.4. A quiver Q without loop is acyclic iff the relation "i ≤ j iff Q contains a path from j to i" defines a partial order on its vertices.

Proof. The relation ≤ is reflexive thanks to the existence of paths of length zero and transitive by concatenation of paths. Now, if i, j are vertices such that i ≤ j and j ≤ i, we get by concatenation an oriented cycle containing i and j. This forces i = j under the hypothesis that Q is acyclic. Conversely, since Q contains no loop by hypothesis, any oriented cycle contains at least two vertices i = j. The latter satisfy i ≤ j and j ≤ i, which contradicts the antisymmetry of the order.

Definition 3.5. A grading of a quiver Q = (Q V , Q A , s, t) is an injection gr : Q V → Z such that for every path π of positive length, gr • s(π) > gr • t(π). An arrow a ∈ Q A gets then graded by the difference gr • s(a) -gr • t(a). Lemma 3.6. A quiver is gradable iff it is acyclic. Proof. Let Q = (Q V , Q A ,
s, t) be a quiver. If it is gradable with grading gr : Q V → Z, then the relation "i ≤ j iff gr(i) ≤ gr(j)" defines a total order on Q V which strictly decreases along paths of positive length, so that Q cannot contain any oriented cycle. Conversely, if Q is acyclic, then the relation "i ≤ j iff Q contains a path from j to i" defines a partial order on Q V by Lemma 3.4. We then get a grading on Q by sending any minimal element i 0 of Q V to zero and then by induction, for every j ≥ 0, by sending any minimal element

i j+1 of Q V \ {i 0 , . . . , i j } to j + 1 ∈ Z. Example 3.7. Let V be a finite subset of Z. Then, the quiver Q = (Q V , Q A , s, t) given by Example 3.2, where Q A contains exactly one arrow for each pair (i, j) ∈ Q 2 V such that i ≤ j, is graded by the inclusion V → Z.

Shellability of a Morse tiling

Recall that a finite relative simplicial complex is the complement S = K \ L for a subcomplex L of a finite simplicial complex K, see [START_REF] Stanley | Generalized H-vectors, intersection cohomology of toric varieties, and related results[END_REF][START_REF] Stanley | Combinatorics and commutative algebra[END_REF][START_REF] Codenotti | On f -and h-vectors of relative simplicial complexes[END_REF]. We may assume that L does not contain any maximal simplex of K, deleting them from K otherwise, and call with some abuse K = S the underlying simplicial complex of S. Finite simplicial complexes themselves are special cases of relative ones for which L is empty. Any Morse tile given by Definition 2.1 is another relative simplicial complex. We now recall the notion of Morse tilings on such finite relative simplicial complexes, as defined in [START_REF] Salepci | Morse shellings and compatible discrete Morse functions[END_REF], see also [START_REF] Salepci | Tilings, packings and expected Betti numbers in simplicial complexes[END_REF][START_REF] Welschinger | Morse shellings on products[END_REF]. Definition 3.8 (Definition 2.8 of [START_REF] Salepci | Morse shellings and compatible discrete Morse functions[END_REF] and 2.5 of [START_REF] Welschinger | Morse shellings on products[END_REF]). A finite relative simplicial complex S = K \ L is Morse tileable iff it admits a partition by Morse tiles such that for every j ≥ 0, the union of tiles of dimension greater than j is closed in S. Such a partition is called a Morse tiling and the collection of simplices underlying these tiles, together with their faces, is the underlying simplicial complex S ⊂ K of S.

Recall that a pure dimensional finite simplicial complex K is classically said to be shellable whenever its maximal simplices can be numbered σ 1 , . . . , σ N in a such way that for every p ∈ {1, . . . , N -1}, σ p+1 \ (σ 1 ∪ • • • ∪ σ p ) is a non-empty union of codimension one faces of σ p+1 , see [START_REF] Bing | Some aspects of the topology of 3-manifolds related to the Poincaré conjecture[END_REF][START_REF] Björner | Shellable and Cohen-Macaulay partially ordered sets[END_REF][START_REF] Kozlov | Combinatorial algebraic topology[END_REF][START_REF] Ziegler | Lectures on polytopes[END_REF]. This condition has been relaxed in several ways, including collapsibility [START_REF] Whitehead | Simplicial Spaces, Nuclei and m-Groups[END_REF][START_REF] Adiprasito | Barycentric subdivisions of convex complexes are collapsible[END_REF][START_REF] Adiprasito | Subdivisions, shellability, and collapsibility of products[END_REF], constructibility [START_REF] Hachimori | Nonconstructible simplicial balls and a way of testing constructibility[END_REF], local constructibility [START_REF] Benedetti | On locally constructible spheres and balls[END_REF][START_REF] Hachimori | Decompositons of simplicial balls and spheres with knots consisting of few edges[END_REF] or partitionability [START_REF] Stanley | Generalized H-vectors, intersection cohomology of toric varieties, and related results[END_REF][START_REF] Stanley | Combinatorics and commutative algebra[END_REF]. Any shelling or partitioning of a pure dimensional finite simplicial complex provides a Morse tiling, where the tiles

σ p+1 \ (σ 1 ∪ • • • ∪ σ p ) are basic, see Example 3.10
for further examples. Likewise, a special class of Morse tilings consists of the following shellable ones and our purpose in this section is to characterize when a tiling is shellable, proving Theorem 1.1. Definition 3.9 (Definition 2.14 of [START_REF] Salepci | Morse shellings and compatible discrete Morse functions[END_REF] and 2.10 of [START_REF] Welschinger | Morse shellings on products[END_REF]). A Morse tiled finite relative simplicial complex S is shellable iff it admits a filtration ∅ = S 0 ⊂ S 1 ⊂ • • • ⊂ S N = S by closed subsets such that for every j ∈ {1, . . . , N }, S j \ S j-1 consists of a single tile.

When the tiling uses only basic tiles, this notion of shelling recovers, in fact slightly extends, the classical one, see Theorem 2.15 and Remark 2.16 of [START_REF] Salepci | Morse shellings and compatible discrete Morse functions[END_REF].

Example 3.10. 1) The boundary of every convex simplicial polytope is shellable by [START_REF] Bruggesser | Shellable decompositions of cells and spheres[END_REF], so that it is Morse shellable, the shelling using only basic tiles.

2) The union of two simplices sharing a common face of codimension greater than one in both simplices is Morse shellable and any Morse shelling uses a closed simplex together with a regular Morse tile of order zero.

3) By Theorem 1.3 of [START_REF] Salepci | Morse shellings and compatible discrete Morse functions[END_REF], every closed triangulated surface is Morse shellable. 4) By Theorem 1.3 of [START_REF] Welschinger | Shellable tilings on relative simplicial complexes and their hvectors[END_REF], every finite simplicial complex becomes Morse shellable after finitely many stellar subdivisions at maximal simplices, or also after a single barycentric subdivision. Figure 2 shows an example of such a Morse shelling taken out from [START_REF] Welschinger | Shellable tilings on relative simplicial complexes and their hvectors[END_REF], where the numbers refer to the indices of its critical tiles. More examples of Morse shellings can be found in [START_REF] Welschinger | Morse shellings on products[END_REF]. We now observe that a Morse tiling supports a quiver which encodes the obstruction to shell it. In particular, the label of the target of an arrow cannot exceed the label of the arrow itself by more than one, see §2.

Example 3.12. 1) If S = T 0 • • • T n+1 is the shelling of ∂∆ n+1 using one n-dimensional basic tile of each order, where ∆ n+1 denotes the standard (n + 1)-simplex, then its quiver is the comparability graph with vertices T 0 , . . . , T n+1 and arrows T j → T i for every 0 ≤ i ≤ j ≤ n + 1, see Corollary 4.2 of [START_REF] Salepci | Tilings, packings and expected Betti numbers in simplicial complexes[END_REF]. 2) If S = T 1 T 1 T 1 is the non-shellable tiling of ∂∆ 2 using three basic tiles of order and dimension one, then its quiver is the oriented boundary of a triangle, see Figure 3.

We now prove Theorem 1.1.

Proof of Theorem 1.1. Let us first assume that a Morse tiled finite relative simplicial complex S is shelled and denote by T 1 , . . . , T N the shelling order on its tiles. Let a : T j → T i be an arrow of its quiver, so that T j ∩ T i = ∅. Since by definition the union ∪ k≤j T k is closed in S, this forces i ≤ j so that the shelling order of S induces a grading of its quiver by Definition 3.5. We deduce from Lemma 3.6 that the latter is acyclic.

Conversely, if the quiver of S is acyclic, it is gradable by Lemma 3.6 and any grading provides a total order on the tiles of S. We may then label them T 1 , . . . , T N in increasing order. Then, for every k ∈ {1, . . . , N } and every 1 ≤ i ≤ k < j ≤ N , T i ∩ T j = ∅, for otherwise the quiver of S would contain an arrow T i → T j by Definition 3.11 which is impossible by Definition 3.5. We deduce that S k = ∪ i≤k T i is closed in S, so that the filtration (S k ) k∈{1,...,N } is a Morse shelling of S. Remark 3.13. 1) The acyclicity condition in Theorem 1.1 may be compared to the criterium given in Theorem 9.3 of [START_REF] Forman | Morse theory for cell complexes[END_REF], up to which a discrete vector field on a simplicial complex is the gradient vector field of a discrete Morse function iff it contains no nonstationary closed path. Indeed, a Morse tiling on a finite simplicial complex encodes a class of compatible discrete vector fields and in the case of a shellable one, these are gradient vector fields of discrete Morse functions whose critical points are in one-to-one correspondance with the critical tiles, preserving the indices, see [START_REF] Salepci | Morse shellings and compatible discrete Morse functions[END_REF] and §5.

2) Every product of a sphere with a torus of positive dimension carries Morse tiled triangulations using only basic tiles, see [START_REF] Welschinger | Morse shellings on products[END_REF]. These have been obtained by subdividing the product of the shelled boundary of a simplex with several copies of the tiling given by Figure 3. These tilings cannot be shelled, so that their quivers all contain oriented cycles, as in the second part of Example 3.12.

We also deduce from Theorem 1.1 criteria of partial shellability of a Morse tiling. Definition 3.14. The q-quiver of a Morse tiled finite relative simplicial complex S has the tiles of order ≤ q + 1 of S as vertices and contains an arrow T → T iff T ∩ T ∩ S (q) = ∅.

The q-skeleton of a Morse tiled finite relative simplicial complex S = K \ L is the finite relative simplicial complex S (q) = K (q) \ L (q) , that is the intersection with S of the q-skeleton of its underlying simplicial complex S ⊂ K.

Theorem 3.15. The q-skeleton S (q) of a Morse tiled finite relative simplicial complex S with acyclic q-quiver is Morse shellable and carries a filtration ∅ = S 0 ⊂ S 1 ∩ S (q) ⊂ • • • ⊂ S N ∩ S (q) = S (q) by closed subset such that for every j ∈ {1, . . . , N }, S j \ S j-1 consists of a single tile of order ≤ q + 1 of S.

Any filtration given by Theorem 3.15 is called a partial shelling. Let us first recall that skeletons of Morse tiles are Morse shellable, see Lemma 2.17 and Theorem 2.18 of [START_REF] Salepci | Morse shellings and compatible discrete Morse functions[END_REF]. Lemma 3.16 (Lemma 2.17 of [START_REF] Salepci | Morse shellings and compatible discrete Morse functions[END_REF]). Skeletons of Morse tiles are Morse shellable.

Proof. Let us first prove the result for codimension one skeletons of Morse tiles. Let T = σ \ (σ 0 ∪ • • • ∪ σ k-1 ∪ µ) be an n-dimensional Morse tile of order k, where σ denotes an n-simplex, σ 0 , . . . , σ n its codimension one faces and µ some higher codimensional face which contains the vertices opposite to σ 0 , . . . , σ k-1 and which can be assumed not to contain the vertex opposite to σ k . Then, ∂σ gets shelled by the tiles T 0 • • • T n , where T 0 = σ 0 and for every i ∈ {1, . . . , n},

T i = σ i \ (σ 0 ∪ • • • ∪ σ i-1
). Likewise, the (n -1)-skeleton of T gets shelled by the Morse tiles (T k \ µ) T k+1 • • • T n , proving the result in this case. Now, replacing T with its (n -1)-skeleton, we deduce that its codimension two skeleton is Morse shellable, a shelling being obtained by concatenation of the shellings of the codimension one skeletons of each of the tiles T k \ µ, T k+1 , . . . , T n . The general case then follows by finite induction.

Proof of Theorem 3.15. Let S be a Morse tiled finite relative simplicial complex with acyclic q-quiver Q and let S be the union of tiles of order ≤ q + 1 of S. Then, S contains the q-skeleton of S since a tile of order greater than q + 1 does not contain any open face of dimension ≤ q, see §2. By Definition 3.14, the vertices of Q are the tiles of S and it contains an arrow T → T iff T ∩ T ∩ S (q) is non-empty. By Lemma 3.6, it is gradable and any grading provides a total order on the tiles of S, so that we may label them T 1 , . . . , T N in increasing order. Then, for every 1 ≤ i ≤ k < j ≤ N , T i ∩ T j ∩ S (q) = ∅, for otherwise Q would contain an arrow T i → T j which is impossible by Definition 3.5. We deduce that S (q) k = ∪ i≤k T i ∩ S (q) is closed in S (q) , so that it provides the desired filtration by closed subsets of S (q) . Now, by Lemma 3.16, the q-skeleton of every Morse tile is Morse shellable. Choosing such a Morse shelling on each T i ∩ S (q) , i ∈ {1, . . . , N }, we get the result by concatenation of these shelling orders.

Spectral sequences

We now observe that any Morse shelling on a finite relative simplicial complex provides a way to compute its relative (co)homology via two spectral sequences. Theorem 4.1. Let S = K \ L be a Morse tiled finite relative simplicial complex having acyclic q-quiver. Then, any partial shelling given by Theorem 3.15 induces a spectral sequence which converges to the relative homology H * (K (q) , L (q) ; G) (resp. relative cohomology H * (K (q) , L (q) ; G)) of its q-skeleton with coefficients in any Abelian group G and whose first page in gradings less than q is the relative homology (resp. cohomology) of its critical tiles. By Proposition 2.4, only the critical tiles of indices less than q contribute to the spectral sequences given by Theorem 4.1 in gradings less than q.

Remark 4.2. The inclusion i of the pair of q-skeletons (K (q) , L (q) ) into the ambient pair of simplicial complexes (K, L) induces an homomorphism i * : H * (K (q) , L (q) ; G) → H * (K, L; G) (resp. i * : H * (K, L; G) → H * (K (q) , L (q) ; G)) in homology (resp. cohomology) which is surjective (resp. injective) in degree q and bijective in degrees less than q, see [START_REF] Munkres | Elements of algebraic topology[END_REF], so that the spectral sequences given by Theorem 4.1 converge to the relative (co)homology of the relative simplicial complex in degrees less than q.

Proof. We may assume that the finite simplicial complex K coincides with S ⊂ K, so that the subcomplex L does not contain any maximal simplex of K. Then, by Theorem 3.15, the q-skeleton of S inherits from any partial shelling a filtration ∅ = S 0 ⊂ S 1 ∩ S (q) ⊂ • • • ⊂ S N ∩ S (q) = S (q) by closed subsets such that for every p ∈ {1, . . . , N }, S p \ S p-1 = T p ∩ S (q) where T p is a tile of order ≤ q + 1 of S. This filtration induces a filtration 0 q) , L (q) ; G) of simplicial chain complexes, where for every p ∈ {1, . . . , N }, K p = S p ∩ K (q) and L p = S p ∩ L (q) , and thus a spectral sequence which converges to the relative homology of the pair (K (q) , L (q) ), see chapter XV of [START_REF] Cartan | Homological algebra[END_REF] for instance. The zero-th page of this spectral sequence is by definition E 0 p, * -p = C * (K p , K p-1 ∪ L p ), p ∈ {1, . . . , N }, where we omit from now on to mention the group of coefficients G. Its first page is E 1 p, * -p = H * (K p , K p-1 ∪ L p ), p ∈ {1, . . . , N } and its limit term is the relative homology of (K (q) , L (q) ) under the form E ∞ p, * -p = H * (K (q) , L (q) ) p /H * (K (q) , L (q) ) p-1 , p ∈ {1, . . . , N }, where H * (K (q) , L (q) ) p denotes the image of the inclusion homomorphism H * (K p , L p ) → H * (K (q) , L (q) ). These images indeed provide the induced filtration 0 ⊂ H * (K (q) , L (q) ) 1 ⊂ • • • ⊂ H * (K (q) , L (q) ) N = H * (K (q) , L (q) ) of the total relative homology of the q-skeleton. Moreover, the filtration being of finite length N , the spectral sequence degenerates at the N -th page, so that E ∞ = E N . Now, for every p ∈ {1, . . . , N }, we deduce by excision an isomorphism H * (K p , K p-1 ∪ L p ) ∼ = H * (T p ∩ K (q) , (T p \T p )∩K (q) ), see [START_REF] Munkres | Elements of algebraic topology[END_REF], and then by composition with the inclusion homomorphism, an isomorphism

⊂ C * (K 1 , L 1 , G) ⊂ • • • ⊂ C * (K N , L N , G) = C * (K (
H * (K p , K p-1 ∪ L p ) ∼ = H * (T p , T p \ T p ) as long as * < q, see Remark 4.2.
The homology part of the statement then follows from Proposition 2.4.

The proof of the cohomology part is similar and obtained from the filtration 0 ⊂ q) , L (q) ; G) of cochain complexes. It induces a spectral sequence which converges to the relative cohomology of the pair (K (q) , L (q) ) and whose zero-th page is by definition E p, * -p

C * (K N , K N -1 ∪ L N ; G) ⊂ • • • ⊂ C * (K N , K 0 ∪ L N ; G) = C * (K (
0 = C * (K N , K p ∪ L N )/C * (K N , K p+1 ∪ L N ) = C * (K p+1 , K p ∪ L p+1 ), p ∈ {0, . . . , N -1}
, where the last isomorphism is given by restriction of the cochains to K p+1 and provides the short exact sequence of the triad (K N , K p+1 ∪ L N , K p ∪ L N ), and where we again omit to mention the coefficient group G from now on. Its first page is thus E p, * -p 1 = H * (K p+1 , K p ∪ L p+1 ), p ∈ {0, . . . , N -1} and its limit term is the relative cohomology of (K (q) , L (q) ) under the form E p, * -p ∞ = H * (K (q) , L (q) ) p /H * (K (q) , L (q) ) p+1 , p ∈ {0, . . . , N -1}, where H * (K (q) , L (q) ) p denotes the image of the inclusion homomorphism H * (K (q) , K p ∪ L (q) ) → H * (K (q) , L (q) ). Now as before, we deduce by excision and Remark 4.2, for every p ∈ {0, . . . , N -1}, an isomorphism H * (K p+1 , K p ∪ L p+1 ) ∼ = H * (T p+1 , T p+1 \ T p+1 ) as long as * < q, so that the result follows from Proposition 2.4.

We now deduce.

Proof of Theorem 1.2. Theorem 1.2 follows from Theorem 4.1 combined with Theorem 1.1 and Proposition 2.4.

Example 4.3. 1) Any shelling of the boundary of a convex simplicial polytope given by [START_REF] Bruggesser | Shellable decompositions of cells and spheres[END_REF] provides a Morse shelling which uses only basic tiles among which a single closed simplex and a single open one. The spectral sequences given by Theorem 1.2 degenerate at the first page and compute the (co)homology of a sphere.

2) The Morse shellings of any closed connected triangulated surface given by Theorem 1.3 of [START_REF] Salepci | Morse shellings and compatible discrete Morse functions[END_REF] can also be chosen to use a single closed simplex by construction, see Theorem 4.1 of [START_REF] Welschinger | Shellable tilings on relative simplicial complexes and their hvectors[END_REF]. Then, the spectral sequences given by Theorem 1.2 again degenerate at the first page whenever the Morse shelling uses a single open simplex as well and they compute then the (co)homology of the surface as the number of critical tiles of each index it uses.

3) In the case of the Morse shelled octahedron depicted in Figure 4 -taken out from [START_REF] Welschinger | Morse shellings on products[END_REF]-, where the numbers appearing in Figure 4 indicate the shelling order, the spectral sequences given by Theorem 1.2 degenerate at the second page. Indeed, the (co)homology classes of the pair of critical tiles with shelling orders 6 and 7, which have consecutive indices one and two, get killed already at this second page. 1 that the k-th Betti number of a Morse tiled simplicial complex with acyclic q-quiver, q > k, is bounded from above by the number of critical tiles of index k of the tiling. In the case of a Morse shellable complex, this recovers Corollary 1.5 of [START_REF] Salepci | Morse shellings and compatible discrete Morse functions[END_REF], which has been deduced from discrete Morse theory.

2) Theorem 1.2 provides a way to compute the (co)homology of a Morse shellable simplicial complex using much smaller (co)chain complexes than the simplicial ones, as the discrete Morse complexes introduced by R. Forman would do, see [START_REF] Forman | Morse theory for cell complexes[END_REF]. For instance, performing a large number of barycentric subdivisions on such a simplicial complex increases by a lot the dimensions of its simplicial (co)chain complexes, whereas by Theorem 1.1 of [START_REF] Salepci | Morse shellings and compatible discrete Morse functions[END_REF], see also [START_REF] Welschinger | Shellable tilings on relative simplicial complexes and their hvectors[END_REF], these inherit Morse shellings using a constant number of critical tiles.

Relation with discrete Morse theory

We already proved in [START_REF] Salepci | Morse shellings and compatible discrete Morse functions[END_REF] that every Morse tiling on a finite simplicial complex encodes a class of compatible discrete vector field which, in the case of a shellable one, are all gradient vector fields of discrete Morse functions. Let us now add that such a discrete Morse function can be chosen in such a way that the filtration given by the Morse shelling, which generates the spectral sequences, coincides with a filtration by sublevel sets of the function. Recall that a subcomplex L of a finite relative simplicial complex K is an elementary collapse of K if and only if K \ L consists of a maximal relative simplex of K together with one of its codimension one face which is not contained in any other maximal simplex of K. It is a collapse of K iff it is obtained from K after finitely many elementary collapses, see for instance [START_REF] Adiprasito | Barycentric subdivisions of convex complexes are collapsible[END_REF][START_REF] Whitehead | Simplicial Spaces, Nuclei and m-Groups[END_REF].

Proposition 5.1. A critical Morse tile of index k collapses onto any of its k-dimensional face, whereas a regular Morse tile collapses onto the empty set.

In the case of a critical tile of maximal index, no collapse is needed at all in Proposition 5.1.

Proof. We proceed by induction on the dimension n of the tiles. If n = 1, a closed onesimplex collapses onto any of its vertices and a closed simplex deprived of one vertex collapses onto the empty set. Let T = σ \ (σ 0 ∪ • • • ∪ σ k-1 ∪ µ) be an n-dimensional Morse tile of order k, where σ denotes an n-simplex, σ 0 , . . . , σ n its codimension one faces and µ some higher codimensional face. If T is critical, let θ be any of the remaining k-dimensional faces of σ ; it contains µ. Then, µ contains the vertices opposite to σ 0 , . . . , σ k-1 and θ can be assumed not to contain the vertex opposite to σ k . We deduce that ∂σ gets shelled by the tiles T 0 • • • T n , where T 0 = σ 0 and for every i ∈ {1, . . . , n}, T

i = σ i \ (σ 0 ∪ • • • ∪ σ i-1 ). If T is of order n + 1, it
is an open simplex and no collapse is needed to get Proposition 5.1. Otherwise, depriving T of its open maximal face together with T n , we deduce that it collapses onto (T k \ µ) T k+1 • • • T n-1 . By the induction hypothesis, it collapses then onto T k \ µ. Since T is critical, dim(µ) = k -1 so that T k \ µ is critical as well and contains θ by hypothesis. It thus collapses onto θ by the induction hypothesis. If T is regular, µ is empty and we deduce likewise that T collapses onto the empty set. Hence the result.

In the framework of discrete Morse theory [START_REF] Forman | Morse theory for cell complexes[END_REF], a sequence of elementary collapses given by Proposition 5.1 is encoded by some discrete vector field V such that for every collapsed pair of faces σ < τ of the tile T , V (σ) = τ and such that V vanishes on the remaining k-dimensional face in case T is critical of index k. We say that a discrete vector field on a Morse tiled finite simplicial complex is compatible with the tiling if and only if it restricts to each tile of the tiling to such a discrete vector field given by Proposition 5.1, compare [START_REF] Salepci | Morse shellings and compatible discrete Morse functions[END_REF].

Theorem 5.2. Let S = K \ L be a Morse shelled finite relative simplicial complex and ∅ = S 0 ⊂ S 1 ⊂ • • • ⊂ S N = S be the corresponding filtration, where for every j ∈ {1, . . . , N }, S j \ S j-1 is a single tile of the shelling. Then, for every discrete vector field V compatible with the shelling, there exists a discrete Morse function f on K = S whose gradient vector field coincides with V on S and such that for every j ∈ {0, . . . , N }, f -1 (] -∞, j]) = S j ∪ L.

Proof. Let T 1 , . . . , T N be the Morse tiles of the shelling and V be any discrete vector field on S compatible with the shelling. For every j ∈ {1, . . . , N }, let us consider a sequence of elementary collapses given by Proposition 5.1 and encoded by the restriction of V to T j . We denote by (τ j 1 , σ j 1 ), . . . , (τ j k j , σ j k j ) the pairs of collapsed faces of T j in the order of collapses, so that (τ j 1 , σ j 1 ) is collapsed first and (τ j k j , σ j k j ) collapsed at the final step. Let us moreover assume that τ j k j = σ j k j in the case T j is critical, so that V (τ j k j ) = 0 in this case, whereas V (σ j i ) = τ j i in all the other ones. Let the restriction of f to L be any discrete Morse function taking non positive values. We then set, for every j ∈ {1, . . . , N } and every i ∈ {1, . . . , k j }, f (σ j i ) = f (τ j i ) = j -1 + 1 i . By construction, it is a discrete Morse function with gradient vector field V on S and such that for every j ∈ {0, . . . , N }, f -1 (] -∞, j]) = S j ∪ L. Hence the result.

In the case of a finite simplicial complex, the critical faces of the Morse functions given by Theorem 5.2 are in one-to-one correspondence with the critical tiles of the shelling, preserving the index, compare [START_REF] Salepci | Morse shellings and compatible discrete Morse functions[END_REF]. The spectral sequences given by Theorem 1.2 are thus associated to the filtration by sublevel sets of some discrete Morse functions on the simplicial complex. They can be computed using discrete Morse theory in the following way. Any discrete Morse function f given by Theorem 5.2 induces a cellular decomposition of K using one cell for each critical face of f , see §7 of [START_REF] Forman | Morse theory for cell complexes[END_REF]. Each page of the spectral sequence can then be computed using this decomposition. Indeed, each cell of the decomposition provides a (co)cycle representative of the relative (co)homology of the corresponding critical tile and thus a representative of the corresponding (co)homology class in the first page. It remains a (co)cycle representative of this (co)homology class as long as the differential of the corresponding page maps it to a shelling degree where the boundary of the cell contains a simplex, which must then be a critical point of degree one less of the Morse function. The latter sits inside a critical tile of the shelling. Then, whether or not this homology class survives to higher pages depends on whether or not this boundary component can be annihilated by another cell, which is encoded by the discrete Morse complex.

It is however not needed to choose a compatible discrete Morse function to compute the spectral sequences, as the case of the octahedron given in Example 4.3 shows. They can be computed directly from the shelling. Remark 5.3. By Theorem 8.10 of [START_REF] Forman | Morse theory for cell complexes[END_REF], the differentials of the Morse complex of a discrete Morse function can also be computed as a finite sum of gradient paths between critical points of consecutive indices. If the Morse function is chosen to be compatible with a Morse shelling, then any such gradient path provides an oriented path in the quiver of the shelling, whose source and target are critical tiles of consecutive indices.

Figure 1 :

 1 Figure 1: The basic tiles in dimension two.

Figure 2 :

 2 Figure 2: A Morse shelled complex in dimension two.

Definition 3 . 11 .

 311 The quiver of a Morse tiled finite relative simplicial complex S has the tiles of S as vertices and contains an arrow T → T iff T ∩ T = ∅. The arrows are labelled by the dimension of the face T ∩ T and the vertices by the order of the corresponding tiles.

Figure 3 :

 3 Figure 3: A non shellable tiling of ∂∆ 2 .

Figure 4 :

 4 Figure 4: A Morse shelling on the octahedron.

Remark 4 . 4 . 1 )

 441 It follows from Theorem 4.