Nermin Salepci

Jean-Yves Welschinger

Tilings, packings and expected Betti numbers in simplicial complexes

Keywords: simplicial complex, tilings, packings, barycentric subdivision, Betti numbers, shellable complex. Mathematics subject classification 2010: 55U10, 60C05, 52C17, 52C22

Let K be a finite simplicial complex. We prove that the normalized expected Betti numbers of a random subcomplex in its d-th barycentric subdivision Sd d (K) converge to universal limits as d grows to +∞. In codimension one, we use canonical filtrations of Sd d (K) to upper estimate these limits and get a monotony theorem which makes it possible to improve these estimates given any packing of disjoint simplices in Sd d (K). We then introduce a notion of tiling of simplicial complexes having the property that skeletons and barycentric subdivisions of tileable simplicial complexes are tileable. This enables us to tackle the problem: How many disjoint simplices can be packed in Sd d (K), d 0?

Introduction 1.Average Betti numbers

Let K be a finite n-dimensional simplicial complex, n > 0, and Sd(K) be its first barycentric subdivision. In [START_REF] Salepci | Asymptotic topology of random subcomplexes in a finite simplicial complex[END_REF] we introduced, for every k ∈ {1, . . . , n}, a construction of codimension k subcomplexes V of Sd(K) parametrized by the (k -1)-dimensional simplicial cochains ∈ C k-1 (K; Z/2Z), see Definition 2.1. The finite set C k-1 (K; Z/2Z) being canonically a probability space, or rather equipped with a family of probability measures (µ ν ) ν∈[0,1] , where µ ν is the product measure for which the probability that the cochain takes the value 0 on a given (k -1)-simplex of K is ν (see [START_REF] Salepci | Asymptotic topology of random subcomplexes in a finite simplicial complex[END_REF]), this construction produces a random variable ∈ C k-1 (K; Z/2Z) → V ⊂ Sd(K) which raises the following questions of random topology: What is the expected topology of V , e.g. its expected Betti numbers? How does this expected topology behave under large iterated barycentric subdivisions? For every p ∈ {0, . . . , n -k}, we denote the p-th Betti number of V by b p (V ) = dim H p (V ; Z/2Z) and its mathematical expectation after d barycentric subdivision by E ν,d (b p ) = C k-1 (Sd d (K);Z/2Z) b p (V )dµ ν ( ), d ≥ 0. We proved in [START_REF] Salepci | Asymptotic topology of random subcomplexes in a finite simplicial complex[END_REF] that there exist universal constants c ± p (n, k), such that

c - p (n, k) ≤ lim inf d→+∞ E ν,d (b p ) (n + 1)! d f n (K) ≤ lim sup d→+∞ E ν,d (b p ) (n + 1)! d f n (K) ≤ c + p (n, k),
where f n (K) denotes the number of n-dimensional simplices of K. These results produced counterparts in this combinatorial framework to the ones obtained in [START_REF] Gayet | Expected topology of random real algebraic submanifolds[END_REF] and [START_REF] Gayet | Universal Components of Random Nodal Sets[END_REF][START_REF] Gayet | Betti numbers of random nodal sets of elliptic pseudodifferential operators[END_REF] on the expected Betti numbers of random real algebraic submanifolds of real projective manifolds or random nodal sets in smooth manifolds respectively (see also [START_REF] Lerario | Statistics on Hilbert's 16th problem[END_REF][START_REF] Nazarov | On the number of nodal domains of random spherical harmonics[END_REF][START_REF] Nazarov | Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions[END_REF]). Our first result is the following:

Theorem 1.1. Let n > 0, k ∈ {1, . . . , n} and p ∈ {0, . . . , n -k}. Then, for every finite n-dimensional simplicial complex K and every ν ∈ [0, 1], the sequence

E ν,d (bp)
fn(K)(n+1)! d d≥0 converges and its limit does not depend on K.

We denote by e p,ν (n, k) the limit given by Theorem 1.1. Recall that with the exception of the case p = 0 (see [START_REF] Nazarov | Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions[END_REF]), the counterpart of Theorem 1.1 in the theory of random polynomials ( [START_REF] Gayet | Expected topology of random real algebraic submanifolds[END_REF]) or random nodal sets ( [START_REF] Nazarov | On the number of nodal domains of random spherical harmonics[END_REF][START_REF] Nazarov | Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions[END_REF][START_REF] Gayet | Betti numbers of random nodal sets of elliptic pseudodifferential operators[END_REF][START_REF] Gayet | Universal Components of Random Nodal Sets[END_REF][START_REF] Lerario | Statistics on Hilbert's 16th problem[END_REF][START_REF] Sarnak | Topologies of nodal sets of random band limited functions[END_REF]) is open. In the case of the standard nsimplex ∆ n , we likewise set bp (V ) = b p (Σ ), where the sum is taken over all connected components Σ of V which do not intersect the boundary of ∆ n . We then set E ν,d ( bp ) = C k-1 (Sd d (∆n);Z/2Z) bp (V )dµ ν ( ) and get: Theorem 1.2. Let n > 0, k ∈ {1, . . . , n} and p ∈ {0, . . . , n -k}. Then, for every ν ∈ [0, 1],

the sequence E ν,d ( bp) (n+1)! d d≥0 is increasing and bounded from above. We denote by ẽp,ν (n, k) the limit of the sequence given by Theorem 1.2. We do not know whether e p,ν (n, k) = ẽp,ν (n, k) or not, except when p = 0. It is related to a problem of percolation which we introduce and discuss in § 2.3, see Theorem 2.7 (or [START_REF] Beffara | Percolation of random nodal lines[END_REF] for another related problem of percolation). Nevertheless we get: Theorem 1.3. Under the hypothesis of Theorem 1.2, e p,ν (n, k) ≥ ẽp,ν (n, k) > 0. Moreover, e 0,ν (n, k) = ẽ0,ν (n, k).

The codimension one case k = 1 plays a special role. Namely, for every ∈ C 0 (K; Z/2Z), we alternatively define V ⊂ |K| such that for every simplex σ ∈ K, V ∩ σ is the convex hull of the middle points of the edges of σ where is not constant, that is where the exterior derivative d does not vanish. We proved in [START_REF] Salepci | Asymptotic topology of random subcomplexes in a finite simplicial complex[END_REF] that the pairs (K, V ) and (K, V ) are always homeomorphic, in fact isotopic, see Proposition 2.2 of [START_REF] Salepci | Asymptotic topology of random subcomplexes in a finite simplicial complex[END_REF]. When K is the moment polytope of some toric manifold equipped with a convex triangulation, the hypersurfaces V coincide with the patchwork hypersurfaces introduced by O. Viro (see [START_REF] Viro | Dequantization of real algebraic geometry on logarithmic paper[END_REF][START_REF] Viro | Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7[END_REF] and Remark 2.4 of [START_REF] Salepci | Asymptotic topology of random subcomplexes in a finite simplicial complex[END_REF]). These hypersurfaces V inherit the structure of a CW complex, having a p-cell for every (p + 1)-simplex σ of K on which is not constant, p ∈ {0, . . . , n -1}, see Corollary 2.5 of [START_REF] Salepci | Asymptotic topology of random subcomplexes in a finite simplicial complex[END_REF]. We denote by C * (V ; Z/2Z) the cellular chain complex of V . Moreover, the p-cell V ∩ σ is isomorphic to a product of two simplices: the simplex spanned by vertices of σ where = 0 and the simplex spanned by vertices of σ where = 1.

We now observe that each ∈ C 0 (K; Z/2Z) induces a filtration ∅ ⊂ K 0 ⊂ K 1 ⊂ . . . ⊂ K [ n+1 2 ] = K, where for every i ∈ {0, . . . , [ n+1 2 ]},

K i = {σ ∈ K | # -1 |σ (0) ≤ i or # -1 |σ (1) ≤ i}.
This i-th subcomplex K i of K is thus the union of all simplices that V meets along the empty set if i = 0 or along a product of two simplices, one of which being of dimension ≤ i -1 if i > 0. Then:

We set, for every p ∈ {0, . . . , n-1} and d ≥ 0, E ν,d (b p (K 0 )) = C 0 (Sd d (K);Z/2Z) b p (K 0 )dµ ν ( ) and will write E ν instead of E ν,0 for simplicity. We deduce: Corollary 1.5. For every finite n-dimensional simplicial complex K, n > 0, and every p ∈ {0, . . . , n -1} and ν ∈

[0, 1], E ν (b p (K 0 )) -b p (K) ≤ E ν (b p (V )) ≤ E ν (b p (K 0 )) + b p+1 (K).
In particular, (n, 1). This result makes it possible to improve the upper estimates of E ν,d (b p ) or of e p,ν (n, k) given in [START_REF] Salepci | Asymptotic topology of random subcomplexes in a finite simplicial complex[END_REF] in the case k = 1 and to relate them with a packing problem in K or Sd d (K), d > 0. To this end, for every finite simplicial complex K, every p ∈ N and ν ∈ [0, 1], we set

lim d→+∞ E ν,d (b p (K 0 )) f n (K)(n + 1)! d = e p,ν
M p,ν (K) = (ν p+1 + (1 -ν) p+1 )b p (K) + ν(1 -ν)(ν p + (1 -ν) p ) dim K i=p+1 (-1) i+1-p (f i (K) -b i (K)),
where f i (K) denotes the number of i-dimensional simplices of K and the sum vanishes if p ≥ dim K. The positivity of M p,ν (K) is closely related to the Morse inequalities associated to the simplicial chain complex of K. Our key result is then the following monotony theorem, (see Theorem 3.7).

Theorem 1.6. Let p ∈ N and ν ∈ [0, 1]. For every finite simplicial complex K and every subcomplex

L of K, 0 ≤ M p,ν (L) -E ν (b p (L 0 )) ≤ M p,ν (K) -E ν (b p (K 0 )).
When L = ∅, Theorem 1.6 combined with Corollary 1.5 already improves the upper estimates given in Corollary 4.2 of [START_REF] Salepci | Asymptotic topology of random subcomplexes in a finite simplicial complex[END_REF] for k = 1. But these get improved further whenever the left hand side in Theorem 1.6 is positive. This turns out to be the case when L is a packing of disjoint simplices in K, or more generally of simplices which intersect along faces of dimensions less than p -1, since E ν (b p (L 0 )) vanishes in this case as soon as p > 0, see Proposition 3.6. Theorem 1.6 combined with Proposition 3.6 thus raises the following packing problem which we tackle in the second part of the paper, independent from the first one: How many disjoint simplices can be packed in the finite simplicial complex K? What about the asymptotic of such a maximal packing in Sd d (K), d 0? Indeed, let us denote by L n,p d the finite set of packings of simplices in Sd d (∆ n ) which intersect each other and the boundary of Sd d (∆ n ) along faces of dimensions less than p-1, where n, d > 0 and p ∈ {0, . . . , n-1}. We set

λ d p,ν (n) = 1 (n+1)! d max L∈L n,p d M p,ν (
L). This sequence (λ d p,ν (n)) d≥0 is increasing and bounded from above, see Proposition 3.9, and we denote by λ p,ν (n) its limit as d grows to +∞. We deduce the following asymptotic result.

Theorem 1.7. For every n > 0, ν ∈ [0, 1] and p ∈ {1, . . . , n -1},

e p,ν (n, 1) ≤ ν(1 -ν)(ν p + (1 -ν) p ) n l=p+1 (-1) l+1-p q l,n -λ p,ν (n).
In Theorem 1.7, q l,n denotes the asymptotic face number lim d→+∞ f l (Sd d (∆n)) (n+1)! d , see [START_REF] Brenti | f -vectors of barycentric subdivisions[END_REF][START_REF] Delucchi | Face vectors of subdivided simplicial complexes[END_REF][START_REF] Salepci | Asymptotic measures and links in simplicial complexes[END_REF]. We do not know the actual value of λ p,ν (n), but the results of the second part of this paper make it possible to estimate λ p,ν (n) from below, see Theorem 1.13. Is it possible to likewise improve the upper estimates of [START_REF] Gayet | Betti numbers of random real hypersurfaces and determinants of random symmetric matrices[END_REF][START_REF] Gayet | Universal Components of Random Nodal Sets[END_REF]? What would then play the role of these packings?

Tilings and Packings

For every positive dimension n and every s ∈ {0, . . . , n + 1}, we define the tile T n s to be the complement of s facets in the standard n-simplex ∆ n . In particular, T n 0 = ∆ n and T n n+1 =

• ∆n, the interior of ∆ n . An n-dimensional simplicial complex K is called tileable when |K| can be covered by disjoint n-dimensional tiles. For instance, the boundary ∂∆ n+1 of the standard (n + 1)-simplex is tileable and it has a tiling which uses each tile T n s exactly once, s ∈ {0, . . . , n + 1}, see Corollary 4.2. The h-vector h(T ) = (h 0 (T ), . . . , h n+1 (T )) of a finite tiling T encodes the number of times h s (T ) each tile T n s is used in the tiling, s ∈ {0, 1, . . . , n + 1}. We observe the following (see Theorem 4.9): Theorem 1.8. Let K be a tileable finite n-dimensional simplicial complex. Then, two tilings T and T of K have the same h-vector provided h 0 (T ) = h 0 (T ). When h 0 (T ) = 1, it coincides with the h-vector of K.

Theorem 1.8 thus provides a geometric interpretation of the h-vector for tileable finite n-dimensional simplicial complexes which have a tiling T such that h 0 (T ) = 1. When K is connected, we call such a tiling regular. Note that such an interpretation was known for shellable simplicial complexes, a closely related notion, see § 4.2. Tileable simplicial complexes have the following key property (see Proposition 4.4 and Corollary 4.15). Theorem 1.9. Let K be a tileable n-dimensional simplicial complex. Then, all its skeletons and barycentric subdivisions are tileable. Moreover, any tiling of K induces a tiling on its skeletons and barycentric subdivisions.

We denote by Sd(T ) the tiling of Sd(K) induced by a tiling T of K. It was proved by A. Björner [START_REF] Björner | Shellable and Cohen-Macaulay partially ordered sets[END_REF] that the barycentric subdivision of a shellable simplicial complex is shellable. Our proof seems more geometric. We actually prove that the first barycentric subdivision Sd(T n s ) of each tile T n s is tileable, s ∈ {0, . . . , n + 1}, see Theorem 4.13. We then study the matrix H n of size (n + 2) × (n + 2) whose rows are the h-vectors of the tilings of Sd(T n s ), s ∈ {0, . . . , n + 1}. Let ρ n be the involution (h 0 , . . . , h n+1 ) ∈ R n+2 → (h n+1 , . . . , h 1 , h 0 ) ∈ R n+2 . We prove the following: Theorem 1.10. For every n > 0, H n is diagonalizable with eigenvalues s!, s ∈ {0, . . . , n+1}. Moreover, it commutes with ρ n and the restriction of ρ n to the eigenspace of s! is (-1) n+1-s id.

We denote by h n = (h n 0 , . . . , h n n+1 ) the eigenvector of the transposed matrix H t n associated to the eigenvalue (n + 1)! and normalize it in such a way that |h n | = n+1 s=0 h n s = 1. We prove that h n 0 = h n n+1 = 0 and that ρ n (h n ) = h n , see Corollary 4.18. Moreover:

Theorem 1.11. Let K be a finite n-dimensional simplicial complex equipped with a tiling T . Then, the sequence

1 |h(T )|(n+1)! d h(Sd d (T )
) converges to h n as d grows to +∞. Moreover, the matrix

1 (n+1)! d H d n converges to (1, . . . , 1)(h n ) t as d grows to +∞.
Hence, the asymptotic h-vector of a tiled finite n-dimensional simplicial complex K does not depend on K and equals h n . This asymptotic result for h-vectors has to be compared with the asymptotic of the f -vectors obtained in [START_REF] Brenti | f -vectors of barycentric subdivisions[END_REF] (see also [START_REF] Delucchi | Face vectors of subdivided simplicial complexes[END_REF] and [START_REF] Salepci | Asymptotic measures and links in simplicial complexes[END_REF]). This notion of tiling makes it possible to study the packing problem in tileable simplicial complexes or rather in their first barycentric subdivisions. In particular, we prove the following: Theorem 1.12. Let K be a finite n-dimensional simplicial complex equipped with a tiling T . Then, it is possible to pack h 0 (T ) + h 1 (T ) disjoint n-simplices in Sd(K). Moreover, this packing can be completed by h n+1-j (T ) + 2 n-1-j n-1-j s=0 hs(T ) 2 s disjoint j-simplices for every j ∈ {0, . . . , n -1}. Since all barycentric subdivisions of the standard simplex are tileable, we are finally able to deduce in the limit the following lower estimates. Theorem 1.13. For every n ≥ 2 and p ∈ {1, . . . , n -1},

λ p,ν (n) ≥ ν(1 -ν)(ν p + (1 -ν) p ) (n + 1)! h n p +2 p-1 p-1 i=0 h n i 2 i n p + 1 + n+1 j=p+1 h n j +2 j-2 j-2 i=0 h n i 2 i n + p -j p + 1 . 1

References 2 Asymptotic behavior of the expected Betti numbers

In the sequel, we denote by K [i] the set of i-dimensional simplices of a simplicial complex K and by K (i) its i-skeleton. We first prove Theorems 1.2 and 1.3 and then Theorem 1.1. In § 2.3, we study a problem of percolation related to these results.

2.1 Proofs of Theorems 1.2 and 1.3

Given a simplicial complex K, a q-simplex of Sd(K) is of the form [σ 0 , . . . , σq ], where σi denotes the barycenter of the simplex σ i of K, i ∈ {0, . . . , q}, and where σ i is a proper face of σ i+1 , i ∈ {0, . . . , q -1}. The block D(σ) dual to a simplex σ ∈ K is the union of all open simplices [σ 0 , . . . , σq ] of Sd(K) such that σ 0 = σ. The union of closed such simplices is denoted by D(σ), see [START_REF] Munkres | Elements of algebraic topology[END_REF]. We recall the following definition from [START_REF] Salepci | Asymptotic topology of random subcomplexes in a finite simplicial complex[END_REF].

Definition 2.1. Let K be an n-dimensional simplicial complex, n > 0, and let k ∈ {1, . . . , n}.

For every ∈ C k-1 (K; Z/2Z), we denote by V the subcomplex of Sd(K) dual to the cocycle d , where d : C k-1 (K; Z/2Z) → C k (K; Z/2Z) denotes the coboundary operator. Hence, V is the union of the blocks D(σ) dual to the k-simplices σ ∈ K such that d , σ = 0.

Proof of Theorem 1.2. For every m ∈ {0, . . . , d}, we observe that

E ν,d ( bp ) = C k-1 (Sd d (∆n);Z/2Z) bp (V )dµ ν ( ) ≥ C k-1 (Sd d (∆n);Z/2Z) σ∈Sd d-m (∆n) [n] bp (V ∩ σ) dµ( ) = σ∈Sd d-m (∆n) [n] C k-1 (Sd m (σ);Z/2Z) bp (V )dµ( ), as µ ν is a product measure, = (n + 1)! d-m E ν,m ( bp ), as f n (Sd d-m (∆ n )) = (n + 1)! d-m .
The sequence

E ν,d ( bp) (n+1)! d d∈N
is thus increasing. Moreover, it is bounded from above by the converging sequence

fp(Sd d+1 (∆n)) (n+1)! d d∈N
, since V is a subcomplex of Sd d+1 (∆ n ), see [START_REF] Brenti | f -vectors of barycentric subdivisions[END_REF][START_REF] Delucchi | Face vectors of subdivided simplicial complexes[END_REF] or [START_REF] Salepci | Asymptotic measures and links in simplicial complexes[END_REF]. Hence the result.

Remark 2.2. For every m ≥ 1, let C(m) be the finite set of homeomorphism classes of pairs (R n , Σ), where Σ is a closed manifold of dimension n -k embedded in R n by an embedding of complexity m, see Definition 5.2 and § 5.3 of [START_REF] Salepci | Asymptotic topology of random subcomplexes in a finite simplicial complex[END_REF]. With the notations of § 5.3 of [START_REF] Salepci | Asymptotic topology of random subcomplexes in a finite simplicial complex[END_REF], we deduce the following lower estimate.

ẽp,ν (n, k) ≥ E ν,d ( bp) (n+1)! d , for every d ≥ 0 by Theorem 1.2, = d m=1 Σ∈C(m) b p (Σ) E ν,d (N Σ ) (n+1)! d ≥ d m=1
Σ∈C(m) b p (Σ)c Σ , by Theorem 5.11 of [START_REF] Salepci | Asymptotic topology of random subcomplexes in a finite simplicial complex[END_REF]. By taking the limit as d grows to +∞, we deduce that ẽp,ν (n, k) ≥ c - p (n, k) by Definition 5.12 of [START_REF] Salepci | Asymptotic topology of random subcomplexes in a finite simplicial complex[END_REF].

Proof of Theorem 1.3. By definition, for every ∈ C k-1 (Sd d (∆ n ); Z/2Z) and p ∈ {0, . . . , n -k}, bp (V ) ≤ b p (V ) so that after integration over C k-1 (Sd d (∆ n ); Z/2Z), we get E ν,d ( bp ) ≤ E ν,d (b p ). The inequality ẽp,ν (n, k) ≤ e p,ν (n, k) is deduced from Theorems 1.1 and 1.2 by passing to the limits as d tends to +∞, after dividing by (n + 1)! d .

When p = 0, we remark that by definition, for every

∈ C k-1 (Sd d (∆ n ); Z/2Z), b 0 (V ) ≤ b0 (V ) + b 0 (V ∩ ∂∆ n ),
as the number of connected components of V which meet ∂∆ n is bounded from above by b 0 (V ∩ ∂∆ n ). We deduce the inequality

E ν,d (b 0 ) (n + 1)! d ≤ E ν,d ( b0 ) (n + 1)! d + E ν,d (b 0 (V ∩ ∂∆ n )) (n + 1)! d . (1)
However, if k = n, E ν,d (b 0 (V ∩ ∂∆ n )) = 0 and so the result follows. If k < n, [START_REF] Salepci | Asymptotic topology of random subcomplexes in a finite simplicial complex[END_REF].

E ν,d (b 0 (V ∩ ∂∆ n )) = C k-1 (Sd d (∆n);Z/2Z) b 0 (V ∩ ∂∆ n )dµ ν ( ) = C k-1 (Sd d (∂∆n);Z/2Z) b 0 (V )dµ ν ( ), as µ ν is a product measure, = O(n! d ), from Theorem 1.2 of
By letting d tend to +∞, we thus deduce from (1) that e 0,ν (n, k) ≤ ẽ0,ν (n, k). Hence the result.

Remark 2.3. The equality e 0,ν (n, k) = ẽ0,ν (n, k) is similar to [START_REF] Nazarov | Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions[END_REF]. Moreover, Theorem 1.3 combined with Remark 2.2 provides the lower bound e p,ν (n, k) ≥ c - p (n, k) given by Corollary 5.13 of [START_REF] Salepci | Asymptotic topology of random subcomplexes in a finite simplicial complex[END_REF]. Theorem 1.3 raises the following question: Are the limits e p,ν (n, k) and ẽp,ν (n, k) equal in general or not? This question is related to a problem of percolation that we discuss in § 2.3.

Proof of Theorem 1.1

In order to prove Theorem 1.1, we first need several preliminary results. For every m ∈ {0, . . . , d}, we consider

A d,m = σ∈Sd d-m (K) [n] • σ and B d,m = τ ∈Sd m (Sd d-m (K) (n-1) ) D(τ ), so that A d,m , B d,m
are two open sets covering the underlying topological space |K|. For every q-simplex θ = [ θ0 , . . . , θq ] ∈ Sd d+1 (K), let l m (θ) ∈ {-1, . . . , q} be the greatest integer i such that θ i ∈ Sd m (Sd d-m (K) (n-1) ), where l m (θ) = -1 if and only if there is no such integer i. When l m (θ) = -1, we moreover set

r θ : θ \ [ θlm(θ)+1 , . . . , θq ] × [0, 1] → θ \ [ θlm(θ)+1 , . . . , θq ] ((1 -α)y + αz, t) → (1 -tα)y + tαz, where α ∈ [0, 1[, y ∈ [ θ0 , . . . , θlm(θ) ] and z ∈ [ θlm(θ)+1 , . . . , θq ].
Proposition 2.4. Let K be a finite simplicial complex of positive dimension n and let θ = [ θ0 , . . . , θq ] be a q-simplex of Sd d+1 (K), q ∈ {0, . . . , n}, d ≥ 0. Then for every m ∈ {0, . . . , d}, Corollary 2.5. Under the hypothesis of Proposition 2.4, for every m ∈ {0, . . . , d},

1. B d,m ∩ θ = θ \ [ θlm(θ)+1 , . . . , θq ] where [ θlm(θ)+1 , . . . , θq ] = ∅ if l m (θ) = q. 2. If l m (θ) = -1, r θ retracts θ \ [ θlm(θ)+1 , . .
1. B d,m is an open subset of |Sd d+1 (K)| = |K| and Sd d+1 (K) \ B d,m is a subcomplex of Sd d+1 (K).
2. For every subcomplex L of Sd d+1 (K), the retractions (r θ ) θ∈L glue together to define

r L : L ∩ B d,m → L ∩ B d,m . Moreover, r L retracts L ∩ B d,m by deformation onto L ∩ Sd m+1 (Sd d-m (K) (n-1)
).

For every subcomplexes

M < L < Sd d+1 (K), the restriction of r L to M is r M . Proof. From the first part of Proposition 2.4, the complement of B d,m in Sd d+1 (K) is the union of simplices θ such that l m (θ) = -1. It is a subcomplex of Sd d+1 (K),
L ∩ B d,m to L ∩ Sd m+1 (Sd d-m (K) (n-1)
). Finally, the last part of Corollary 2.5 follows from the last part of Proposition 2.4.

Proposition 2.6. Under the hypothesis of Theorem 1.1, there exists a universal constant c(n) such that for every m ∈ {0, . . . , d},

|E ν,d (b p ) -E ν,d (b p (V ∩ A d,m ))| ≤ (n + 1)! d f n (K) c(n) (n + 1) m + O(n! d ).
Proof. It follows from Corollary 2.5 that for every

∈ C k-1 (Sd d (K); Z/2Z), (A d,m ∩ V ) ∪ (B d,m ∩ V
) is an open cover of V . The long exact sequence of Mayer-Vietoris associated to this open cover (see § 33 of [START_REF] Munkres | Elements of algebraic topology[END_REF]) reads

. . . → H p (A d,m ∩B d,m ∩V ) ip -→ H p (A d,m ∩V )⊕H p (B d,m ∩V ) jp -→ H p (V ) ∂p -→ H p-1 (A d,m ∩B d,m ∩V ) → . . .
where the coefficients are in Z/2Z. We thus deduce that

b p (V ) = dim Im(∂ p ) + dim Im(j p ) = dim Im(∂ p ) + b p (A d,m ∩ V ) + b p (B d,m ∩ V ) -dim Im(i p ).
After integration over C k-1 (Sd d (K); Z/2Z), we get

|E ν,d (b p ) -E ν,d (A d,m ∩ V )| ≤ E ν,d (b p (B d,m ∩ V )) + E ν,d (b p (A d,m ∩ B d,m ∩ V )) + E ν,d (b p-1 (A d,m ∩ B d,m ∩ V )). (2)
Let us now bound each term of the right hand side. We know from Corollary 2.5 that for every

∈ C k-1 (Sd d (K); Z/2Z), r V retracts V ∩ B d,m by deformation onto V ∩ Sd m+1 (Sd d-m (K) (n-1) ). Thus, E ν,d (b p (V ∩ B d,m )) = E ν,d b p V ∩ Sd m+1 (Sd d-m (K) (n-1) ) ≤ f p (Sd m+1 (Sd d-m (K) (n-1) )) ≤ n-1 i=p σ∈Sd d-m (K) [i] f p (Sd m+1 (σ)) ≤ f * (Sd d-m (K))f p (Sd m+1 (∆ n-1 )),
where f * (Sd d-m (K)) denotes the total face number of Sd d-m (K).

Let K be the subcomplex of K made of simplices which are faces of n-simplices of K. Then,

f * (Sd d-m (K)) = f * (Sd d-m ( K)) + f * (Sd d-m (K \ K)) ≤ f n (K)f * (Sd d-m (∆ n )) + f * (Sd d-m (K \ K)) ≤ f n (K)f * (Sd d-m (∆ n )) + O(n! d-m ),
from [START_REF] Brenti | f -vectors of barycentric subdivisions[END_REF], see also [START_REF] Delucchi | Face vectors of subdivided simplicial complexes[END_REF][START_REF] Salepci | Asymptotic measures and links in simplicial complexes[END_REF], since the simplices of K \ K are of dimension at most (n-1). Likewise, the sequences f * (Sd d-m (∆n))

(n+1)! d-m d∈N and fp(Sd m+1 (∆ n-1 )) n! m+1 m∈N
are universal and convergent, see [START_REF] Brenti | f -vectors of barycentric subdivisions[END_REF][START_REF] Delucchi | Face vectors of subdivided simplicial complexes[END_REF]. They are thus bounded by universal constants. We deduce the existence of a universal constant

c 1 (n) such that E ν,d (b p (B d,m ∩ V )) ≤ f n (K)(n + 1)! d c 1 (n) (n+1) m + O(n! d ).
Similarly, from Corollary 2.5 we know that for every

∈ C k-1 (Sd d (K); Z/2Z) and every σ ∈ Sd d-m (K) [n] , r V ∩σ is a deformation retract of B d,m ∩ • σ ∩V onto ∂σ ∩ V . So we have E ν,d (b p (A d,m ∩ B d,m ∩ V )) = σ∈Sd d-m (K) [n] E ν,d (b p (B d,m ∩ • σ ∩V )) = σ∈Sd d-m (K) [n] E ν,d (b p (V ∩ ∂σ)) ≤ σ∈Sd d-m (K) [n] f p (Sd m+1 (∂σ)) ≤ f n (K) (n+1)! d (n+1) m fp(Sd m+1 (∂∆n) n! m .
Once again the sequence

fp(Sd m+1 (∂∆n)) n! m m∈N
is universal and convergent and thus gets bounded by a universal constant. The third term in (2) is analogous to the second one, so it gets bounded by a universal constant as well. Hence the result.

Proof of Theorem 1.1. We first prove the result for K = ∆ n and then deduce it for a general simplicial complex K. For every d ≥ 0, we set

u d = 1 (n + 1)! d C k-1 (Sd d (∆n);Z/2Z) b p (V )dµ ν ( ). For every ∈ C k-1 (Sd d (∆ n ); Z/2Z) and every m ∈ {0, . . . , d}, b p (V ∩ A d,m ) = σ∈Sd d-m (∆n) [n] b p (V ∩ • σ). After integration over C k-1 (Sd d (∆ n ); Z/2Z), we get E ν,d (b p (V ∩A d,m )) = (n+1)! d-m E ν,m (b p ) since µ ν is a product measure, #Sd d-m (∆ n ) [n] = (n + 1)! d-m (see [3]) and b p (V ∩ • σ) = b p (V ∩ σ) by Corollary 2.
|u d -u m | ≤ c(n) (n + 1) m + O 1 (n + 1) d .
The sequence (u d ) d∈N is thus a Cauchy sequence and so a converging sequence. The result follows for K = ∆ n . Let us now suppose that K is any finite simplicial complex of dimension n and set, for every d ≥ 0,

v d = 1 f n (K)(n + 1)! d C k-1 (Sd d (K);Z/2Z) b p (V )dµ ν ( ).
We deduce from Proposition 2.6, after dividing by

f n (K)(n + 1)! d , that |v d -u m | ≤ c(n) (n+1) m + O 1 (n+1) d .
Since (u m ) m∈N converges, this implies that (v d ) d∈N converges as well and that lim d→+∞ v d = lim m→+∞ u m . Hence the result.

Percolation

Let n > 0 and k ∈ {1, . . . , n}. For every 0 ≤ m ≤ d and every σ ∈ Sd d-m (∆ n ) [n] , we denote by

P d,m (σ) the set of ∈ C k-1 Sd m (Sd d-m (∆ n )\ • σ)
(P m+m ,m (σ)) = 1, for every σ ∈ Sd m (∆ n ) [n] , m ∈ N.
Proof. If p = 0, e p,ν (n, k) = ẽp,ν (n, k) for every k ∈ {1, . . . , n} and n > 0, by Theorem 1.3. If n = 2, we need to prove the equality for k = p = 1. In this case, for every ∈ C 0 (Sd d (∆ 2 ); Z/2Z), V is a homological manifold of dimension one by Theorem 1.1 of [START_REF] Salepci | Asymptotic topology of random subcomplexes in a finite simplicial complex[END_REF]. The connected components of V which meet ∂∆ 2 are thus homeomorphic to ∆ 1 so that b 1 (V ) = b1 (V ). The equality e 1 (2, 1) = ẽ1 (2, 1) is then obtained by integrating over

C 0 (Sd d (∆ 2 ); Z/2Z).
Let us suppose now that there exists n > 0, k ∈ {1, . . . , n} and p ∈ {0, . . . , n -k} such that e p,ν (n, k) > ẽp,ν (n, k). Let m ∈ N and σ ∈ Sd m (∆ n ) [n] . For every m ≥ 1, we denote by P m +m (σ) the set of ˜ ∈ C k-1 (Sd m +m (∆ n ) \ Sd m (σ)) for which there exists ∈ P m +m (σ) whose restriction to the complement of σ is ˜ (Sd m+m (∆ n ) \ Sd m (σ) is not a simplicial complex. It is enough to show that lim m→+∞ µ ν ( P m+m ,m (σ)) = 1. Indeed, let σ ∈ Sd 2 (σ) which does not meet the boundary of σ. Then, for every m ≥ 1 and ˜ ∈ P m+m +2,m (σ), the restric-

tion of ˜ to Sd m+2 (Sd m (∆ n )\ • σ) lies in P m+m +2,m+2 (
σ), since a path in V connecting ∂∆ n to ∂ σ has to cross ∂σ. Let P m+m +2,m+2 (σ) be the image of P m+m +2,m+2 (σ) in P m+m +2,m+2 (σ) by this restriction map, so that µ ν ( P m+m +2,m+2 (σ)) ≤ µ ν (P m+m +2,m+2 (σ)). Since µ ν is a product measure, µ ν ( P m+m +2,m+2 (σ)) ≥ µ ν ( P m+m +2,m+2 (σ)) and we deduce the inequality µ ν ( P m+m +2,m+2 (σ)) ≤ µ ν (P m+m +2,m+2 (σ)). Thus, the fact that lim m→+∞ µ ν ( P m+m +2,m+2 (σ)) = 1 implies that lim m→+∞ µ ν (P m+m +2,m+2 (σ)) = 1.

Let us now prove that lim m→+∞ µ ν ( P m+m ,m (σ)) = 1, a result which does not depend on the choices of m and σ. For every ∈ C k-1 (Sd m +m (∆ n )), let Λ be the union of the connected components of V which meet ∂∆ n , so that b p (V ) = bp (V ) + b p (Λ ). Replacing V by Λ in Proposition 2.6, we deduce that

lim m→+∞ 1 (n + 1)! m +m |E m +m (b p (Λ )) -E m +m (b p (Λ ∩ A m+m ,m ))| = 0.
Under our hypothesis we then deduce that

lim m→+∞ 1 (n + 1)! m +m E m +m (b p (Λ ∩ A m+m ,m )) = e p,ν (n, k) -ẽp,ν (n, k) > 0.
However, by definition,

E m+m (b p (Λ ∩ A m+m ,m )) = σ∈Sd m (∆n) [n] C k-1 (Sd m+m (∆n)\Sd m (σ)) C k-1 (Sd m (σ)) b p (Λ ∩ • σ)dµ ν ( ) = σ∈Sd m (∆n) [n] P m+m ,m (σ) C k-1 (Sd m (σ)) b p (Λ ∩ • σ)dµ ν ( ) ≤ σ∈Sd m (∆n) [n] µ ν ( P m+m ,m (σ))E ν,m (b p -bp ), so that 1 (n + 1)! m σ∈Sd m (∆n) [n] µ ν ( P m+m ,m (σ)) ≥ E ν,m +m (b p (Λ ∩ A m+m ,m )) (n + 1)! m E ν,m (b p -bp ) .
From what precedes, the right hand side converges to 1 as m grows to +∞. We thus deduce that for every m ∈ N and every σ ∈ Sd m (∆ n ) [n] , lim m→+∞ µ ν ( P m+m ,m (σ)) = 1. Hence the result.

Remark 2.8.

1. It would be interesting to prove that for every m ∈ N, the sequence

1 (n+1)! m σ∈Sd m (∆n) [n] µ ν ( P m+m ,m (σ)
) which appeared at the end of the proof of Theorem 2.7 converges to a limit u m as m grows to +∞, without assuming that e p,ν (n, k) > ẽp,ν (n, k). Then, the sequence (u m ) m ∈N would be submultiplicative.

2. It would also be interesting to know whether the sequence µ ν (P m+m ,m (σ)) m∈N converges or not for every m and every σ ∈ Sd m (∆ n ) [n] , and whether or not the limit is 1 without assuming that e p,ν (n, k) > ẽp,ν (n, k). Nevertheless, if this condition is necessary by Theorem 2.7 for e p,ν (n, k) to be greater than ẽp,ν (n, k), it is not sufficient, a priori. This question of percolation seems already of interest in dimension two.

3 Refined upper bounds

Induced filtrations

Let K be a finite n-dimensional simplicial complex. For every ∈ C 0 (K; Z/2Z), K inherits the filtration

∅ ⊂ K 0 ⊂ K 1 ⊂ . . . ⊂ K [ n+1 2 ] = K, where K i = {σ ∈ K|#( | σ ) -1 (0) ≤ i or #( |σ ) -1 (1) ≤ i}, i ∈ {0, 1, . . . , [ n+1 2
]}. We denote by C * (K, K 0 ; Z/2Z) the associated relative simplicial chain complex and by C * (V ; Z/2Z) the cellular chain complex of the CW-complex V , see Corollary 2.5 of [START_REF] Salepci | Asymptotic topology of random subcomplexes in a finite simplicial complex[END_REF]. We now prove Theorem 1.4 and Corollary 1.5.

Proof of Theorem 1.4. For every p ∈ {0, . . . , n-k}, the relative chain complex C p (K, K 0 ; Z/2Z) contains a canonical basis given by simplices on which is not constant. Similarly, a basis of C p (V ; Z/2Z) is given by a p-cell for every (p + 1)-simplex of K on which is not constant. Hence the canonical isomorphism between C * (K, K 0 ; Z/2Z) and C * (V ; Z/2Z) [1] . Moreover, the boundary operator of C * (V ; Z/2Z) [1] gets identified to the composition pr • ∂, where ∂ is the boundary operator of C * (K; Z/2Z) and pr is the projection on C * (K, K 0 ; Z/2Z).

Theorem 1.4 gives new insights on the following corollary which has been established in [START_REF] Salepci | Asymptotic topology of random subcomplexes in a finite simplicial complex[END_REF], see also [START_REF] Salepci | Asymptotic measures and links in simplicial complexes[END_REF]. Recall that for every finite simplicial complex of dimension n, q K (T ) denotes the face polynomial n p=0 f p (K)T p .

Corollary 3.1 (Theorem 1.5 of [START_REF] Salepci | Asymptotic topology of random subcomplexes in a finite simplicial complex[END_REF]). For every finite n-dimensional simplicial complex K and every ν ∈

[0, 1], χ(K) + E ν (K) = νq K (-ν) + (1 -ν)q K (ν -1).
Proof. For every ∈ C 0 (K; Z/2Z), the short exact sequence

0 → C(K 0 ) i - → C(K) π -→ C(K, K 0 ) → 0.
combined with Theorem 1.4 implies χ(K 0 ) -χ(V ) = χ(K). After integration over all , we get

χ(K) + E(χ) = E(χ(K 0 )) = σ∈K (-1) dim σ µ{ ∈ C 0 (K; Z/2Z)| |σ = constant} = n p=0 (-1)(ν p+1 + (1 -ν) p+1 )f p (K).
Proof of Corollary 1.5. The short exact sequence 0

→ C(K 0 ) i - → C(K) π -→ C(K, K 0 ) → 0 induces the long exact sequence in homology . . . → H p+1 (K) (π p+1 ) * -----→ H p+1 (K, K 0 ) (∂ p+1 ) * -----→ H p (K 0 ) (ip) * ---→ H p (K) → . . . . We deduce, b p+1 (K, K 0 ) = dim ker (∂ p+1 ) * + dim Im(∂ p+1 ) * = dim Im(π p+1 ) * + dim(ker i p ) * = b p+1 (K) + b p (K 0 ) -dim Im(i p+1 ) * -dim Im(i p ) * ≤ b p+1 (K) + b p (K 0 ).
The upper estimates follow using the isomorphism H p+1 (K, K 0 ) → Hp (V ) given by Theorem 1.4. We furthermore deduce

b p+1 (K, K 0 ) = dim ker (∂ p+1 ) * + b p (K 0 ) -dim Im(i p ) * = dim ker(∂ p+1 ) * + b p (K 0 ) -b p (K) + dim Im(π p ) * ≥ b p (K 0 ) -b p (K).
The isomorphism given by Theorem 1.4 now implies the lower estimate after integration over C 0 (K; Z/2Z). Finally, the asymptotic result follows from the definition of e p,ν (n, 1) and the invariance of the Betti numbers of K under barycentric subdivisions.

Thanks to Corollary 1.5, in order to estimate E ν (b p ) from above, it suffices to upper estimate E ν (b p (K 0 )). A rough upper estimate is given by Proposition 3.2, but it is going to be improved by Theorem 3.4 and improved further by Corollary 3.8. Proposition 3.2. Let K be a finite n-dimensional simplicial complex, n > 0. For every ν ∈ [0, 1] and every p ∈ {0, . . . , n -1},

E ν (b p (K 0 )) ≤ f p (K)(ν p+1 + (1 -ν) p+1 ) = E ν (f p (K 0 )).
Proof. The result follows from the fact that b p (K 0 ) ≤ f p (K 0 ) for every ∈ C 0 (K; Z/2Z), combined with

E(f p (K 0 )) = C 0 (K;Z/2Z) #{σ ∈ K [p] | |σ is constant}dµ( ) = σK [p] µ{ ∈ C 0 (K; Z/2Z)| |σ is constant} = f p (K)(ν p+1 + (1 -ν) p+1 ).
Lemma 3.3. Let K be a finite n-dimensional simplicial complex, n ≥ 0. For every p ∈ {0, . . . , n}, the space of p-cycles Z p (K) ⊂ C p (K) has a complement spanned by simplices of K.

Proof. We proceed by induction on the codimension of Z p (K). If Z p (K) contains all the simplices, there is nothing to prove. Otherwise, we choose a simplex σ / ∈ Z p (K), so that σ ∩ Z p (K) = {0}. Then we consider the quotient C p (K)/ σ and the image of Z p (K) in this quotient. Its codimension decreases by one. The result follows by induction.

The upper estimate given in Propositon 3.2 can now be improved, see § 1.1 for the definition of M p,ν (K).

Theorem 3.4. Let ν ∈ [0, 1] and p ∈ N. Then, for every finite simplicial complex K,

E ν (b p (K 0 )) ≤ M p,ν (K).
Proof. By definition, for every

∈ C 0 (K; Z/2Z), b p (K 0 ) = dim Z p (K 0 ) -dim B p (K 0 ) = f p (K 0 ) -dim B p-1 (K 0 ) -dim B p (K 0 ). Let S p (K) be a subspace complementary to Z p (K) in C p (K). Its dimension equals dim S p (K) = f p (K) -dim Z p (K) = (f p (K) -b p (K)) -dim B p (K) = n i=p (-1) k-p (f i (K) -b i (K))
, by induction.

From Lemma 3.3 we can choose S p (K) to be spanned by simplices. A linear combination of such simplices belongs to C p (K 0 ) if and only if each simplex belongs to C p (K 0 ) = ker pr . Therefore, dim(S p (K) ∩ C p (K 0 )) = #{σ ∈ B p |σ ∈ K 0 }, where B p denotes our basis of S p (K). However,

S p (K) ∩ C p (K 0 ) is transverse to ker ∂ K 0 p in C p (K 0 ), so that dim(S p (K) ∩ C p (K 0 )) ≤ dim B p-1 (K 0 ). We deduce E ν (b p (K 0 )) ≤ E ν (f p (K 0 )) -E ν (#{σ ∈ B p |σ ∈ K 0 }) -E ν (#{σ ∈ B p+1 |σ ∈ K 0 }) ≤ (ν p+1 + (1 -ν) p+1 )(f p (K) -#B p ) -(ν p+2 + (1 -ν) p+2 )#B p+1 ≤ (ν p+1 + (1 -ν) p+1 ) f p (K) -dim K i=p (-1) i-p (f i (K) -b i (K)) - (ν p+2 + (1 -ν) p+2 ) dim K i=p+1 (-1) i+1-p (f i (K) -b i (K)) ≤ (ν p+1 + (1 -ν) p+1 )b p (K) + ν(1 -ν)(ν p + (1 -ν) p ) dim K i=p+1 (-1) i+1-p f i (K) -b i (K).
The third inequality follows from the fact that #B p = dim S p (K).

Monotony theorem

Lemma 3.5. For every n > 0 and p ∈ {1, . . . , n-1}, M p,ν

(∆ n ) = ν(1-ν)(ν p +(1-ν) p ) n p+1 , while M 0,ν (∆ n ) = 1 + ν(1 -ν)2n.
Proof. The simplex ∆ n is contractible, so that b 0 (∆ n ) = 1 and b i (∆ n ) = 0 for every i > 0. Moreover, for every i ∈ {0, . . . , n}, f i (∆ n ) = n+1 i+1 as the i-simplices of ∆ n are in one-to-one correspondence with sets of i + 1 vertices of ∆ n .

Thus, for p ∈ {1, . . . , n -1},

n i=p+1 (-1) i+1-p n+1 i+1 = n-1 i=p+1 (-1) i+1-p ( n i + n i+1 ) + (-1) n+1-p = (-1) n-p + n p+1 + (-1) n+1-p = n p+1
, and the result follows from the definition of M p,ν (∆ n ), see § 1.1.

Let us remark that the left hand side of Theorem 3.4 vanishes for a large family of complexes, for instance those given by the following proposition. Proposition 3.6. Let p > 0 and L be a finite simplicial complex which contains a family of simplices

{σ i ∈ L, i ∈ I} such that 1. ∀τ ∈ L, ∃i ∈ I such that τ < σ i , 2. ∀i = j ∈ I, dim(σ i ∩ σ j ) < p -1.
Then, for every l ≥ p, b p (L) = 0 = E ν (b p (L 0 )).

Proof. Let D = i∈I σ i be the disjoint union of the σ i , i ∈ I, and h : D → L be the associated canonical simplicial map. From the hypothesis, for every l ≥ p -1, h : D → L provides a bijection between the l-simplices of D and L. Thus, h # : C l (D) → C l (L) is an isomorphism of vector spaces and a chain map which induces an isomorphism h * : H l (D) → H l (L). As H l (D) = 0, we deduce that b l (L) vanishes. Now, for every ∈ C 0 (L, Z/2Z), we set ˜ = • h. Then, h : D ˜ 0 → L 0 induces another isomorphism h * : H l (D ˜ 0 ) → H l (L 0 ) for l ≥ p. However, by definition, D ˜ 0 is again a disjoint union of simplices, so that H l (D ˜ 0 ) = 0. Hence the result.

The following monotony theorem completes Theorem 1.6. Theorem 3.7. Let ν ∈ [0, 1] and p ∈ N. For every finite simplicial complex K and every subcomplex

L of K, M p,ν (L) -E ν (b p (L 0 )) ≤ M p,ν (K) -E ν (b p (K 0 )).
Similarly,

E ν (f p (L)) -E ν (b p (L 0 )) ≤ E ν (f p (K)) -E ν (b p (K 0 )).
Proof. Let S p (L) (resp. S p+1 (L)) be complementary to Z p (L) (resp. Z p+1 (L)) in C p (L) (resp. C p+1 (L)) and spanned by p-simplices, see Lemma 3.3. The intersection of S p (L) with

Z p (K) is thus {0} and Z p (K) ⊕ S p (L) = Z p (K) + C p (L). Let S p (K, L) (resp. S p+1 (K, L)) be the complement of this space in C p (K) (resp. in C p+1 (K)) spanned by p-simplices of K in such a way that S p (K, L) completes S p (L) to a complement S p (K) = S p (L) ⊕ S p (K, L) of Z p (K) in C p (K) (resp. S p+1 (K, L) completes S p+1 (L) to a complement S p+1 (K) = S p+1 (L)⊕ S p+1 (K, L) of Z p+1 (K) in C p+1 (K)). Let B p (K) ⊂ C p (K) be the image of ∂ p+1 : C p+1 (K) → C p (K). For every ∈ C 0 (K; Z/2Z), L 0 = K 0 ∩ L, so that B p-1 (L 0 ) ⊂ B p-1 (K 0 ) and B p (L 0 ) ⊂ B p (K 0 ). Moreover, S p (K, L) ∩ C p (K 0 ) is complement to Z p (K 0 ) + C p (L 0 ) in C p (K 0 ), so that dim(S p (K, L) ∩ C p (K 0 )) + dim B p-1 (L 0 ) ≤ dim B p-1 (K 0 ) and similarly dim(S p+1 (K, L) ∩ C p+1 (K 0 )) + dim B p (L 0 ) ≤ dim B p (K 0 ). As before we deduce b p (K 0 ) = f p (K 0 ) -dim B p-1 (K 0 ) -dim B p (K 0 ) ≤ f p (K 0 ) -f p (L 0 ) + f p (L 0 ) -dim B p-1 (L 0 ) -dim B p (L 0 ) - dim(S p (K, L) ∩ C p (K 0 )) -dim(S p+1 (K, L) ∩ C p+1 (K 0 )) ≤ f p (K 0 ) -dim(S p (K) ∩ C p (K 0 )) -dim(S p+1 (K) ∩ C p+1 (K 0 )) + b p (L 0 )- f p (L 0 ) -dim(S p (L) ∩ C p (L 0 )) -dim(S p+1 (L) ∩ C p+1 (L 0 )) .
The first part of the result is thus obtained as in Theorem 3.4 by integrating over C 0 (K; Z/2Z) and the second part follows from the inequality dim S

p (K)∩C p (K 0 ) ≥ dim S p (L)∩ C p (L 0 ) (resp. dim S p+1 (K) ∩ C p+1 (K 0 ) ≥ dim S p+1 (L) ∩ C p+1 (L 0 )).
Thanks to Proposition 3.6, Theorem 3.7 makes it possible to improve the upper bounds given by Theorem 3.4.

Corollary 3.8. Under the hypothesis of Theorem 3.7, let L be subcomplex of K satisfying the hypothesis of Proposition 3.6. Then,

E ν (b p (K 0 )) ≤ (ν p+1 +(1-ν) p+1 )b p (K)+ν(1-ν)(ν p +(1-ν) p ) dim K i=p+1 (-1) i+1-p f i (K)-f i (L)-b i (K) . Proof. From Theorem 3.7, E ν (b p (K 0 )) ≤ M p,ν (K) -M p,ν (L) + E ν (b p (L 0 )) and from Proposi- tion 3.6, E ν (b p (L 0 )) = 0 and M p,ν (L) = ν(1 -ν)(ν p + (1 -ν) p ) n i=p+1 (-1) i+1-p f i (L)
. Hence the result.

For every d > 0, let L n,p d be the finite set of simplicial subcomplexes L of Sd d (∆ n ) containing a family of simplices

{σ i ∈ L, i ∈ I} such that 1. ∀τ ∈ L, ∃i ∈ I such that τ < σ i , 2. ∀i = j ∈ I, dim(σ i ∩ σ j ) < p -1, 3. ∀i ∈ I, dim(σ i ∩ Sd d (∂∆ n )) < p -1. We set λ d p,ν (n) = 1 (n+1)! d max L∈L n,p d M p,ν (L).
Proposition 3.9. The sequence (λ d p,ν (n)) d≥0 is increasing and bounded.

Let λ p,ν (n) be the limit of this sequence (λ d p,ν (n)) d≥0 .

Proof

-→ σ. Let L d+m = σ∈Sd m (∆n) [n] (f σ ) * L d . It is a subcomplex of Sd d+m (∆ n ). Moreover, L d+m belongs to L n,p d+m , so that 1 (n+1)! d+m M p,ν (L d+m ) ≤ λ d+m p,ν (n). From Proposition 3.6, M p,ν (L d+m ) = ν(1 -ν)(ν p + (1 -ν) p ) n i=p+1 (-1) i+1-p f i (L d+m ). However, by construction, L d+m ∩ Sd d (Sd m (∆ n ) (n-1)
) is of dimension < p -1 so that it does not contribute to the computation of M p,ν (L d+m ). As the cardinality of Sd m (∆ n ) [n]

is (n + 1)! m , we deduce that M p,ν (L d+m ) = (n + 1)! m M p,ν (L d ). Thus λ d p,ν (n) ≤ λ d+m p,ν (n). Now, for every L ∈ L n,p d , M p,ν (L) ≤ M p,ν (Sd d (∆ n )) = ν(1 -ν)(ν p + (1 -ν) p ) n i=p+1 (-1) i+1-p f i (Sd d (∆ n )).
The sequence

Mp,ν (Sd d (∆n)) (n+1)! d
is convergent, see [START_REF] Brenti | f -vectors of barycentric subdivisions[END_REF][START_REF] Delucchi | Face vectors of subdivided simplicial complexes[END_REF], so that the sequence λ d p,ν (n) is bounded. Hence the result.

We can now deduce Theorem 1.7. Recall that by definition, for every i ∈ {0, . . . , n}, q i,n = lim d→+∞

f i (Sd d (K)) fn(K)(n+1)! d .
Proof of Theorem 1.7. We proceed as in the proof of Proposition 3.9. For every n-simplex σ ∈ K, we choose a simplicial isomorphism f σ : ∆ n → σ. For every d > 0, there exists a subcomplex L d ∈ L n,p d which maximize the function M p,ν over L n,p d . We set

K d = σ∈K [n] (f σ ) * L d .
We deduce as in the proof of Proposition 3.9 that M p,ν

(K d ) = f n (K)M p,ν (L d ). Since b p+1 (Sd(K)) = b p+1 (K), we deduce E ν,d (b p ) ≤ E ν (b p (Sd d (K) 0 )) + b p+1 (K), from Corollary 1.5, ≤ M p,ν (Sd d (K)) -M p,ν (K d ) + b p+1 (K), from Proposition 3.6 and Theorem 3.7, ≤ M p,ν (Sd d (K)) -f n (K)M p,ν (L d ) + b p+1 (K). However, M p,ν (Sd d (K)) = b p (K)(ν p+1 + (1 -ν) p+1 )+ν(1-ν)(ν p +(1-ν) p ) n i=p+1 (-1) i+1-p (f i (Sd d (K))-b i (K)), so that lim d→+∞ Mp,ν (Sd d (K)) fn(K)(n+1)! d = ν(1 -ν)(ν p + (1 -ν) p ) n
i=p+1 (-1) i+1-p q i,n , see [START_REF] Brenti | f -vectors of barycentric subdivisions[END_REF][START_REF] Delucchi | Face vectors of subdivided simplicial complexes[END_REF] or [START_REF] Salepci | Asymptotic topology of random subcomplexes in a finite simplicial complex[END_REF].

By definition, lim

d→+∞ Mp,ν (L d ) (n+1)! d = λ p,ν (n) and thus lim d→+∞ E d (b p ) f n (K)(n + 1)! d ≤ ν(1 -ν)(ν p + (1 -ν) p ) n i=p+1 (-1) i+1-p q i,n -λ p,ν (n).
In the light of Theorem 1.7, it would be great to be able to compute λ p,ν (n). Theorem 1.13 provides an estimate of this number from below.

Tilings

Tiles

For every n > 0 and every s ∈ {0, 1, . . . , n + 1}, we set T n s = ∆ n \ (σ 1 ∪ . . . ∪ σ s ), where σ i denotes a facet of ∆ n , i ∈ {1, . . . , s}. In particular, the tile T n n+1 is the open n-simplex

f j (T n s ) = 0 if 0 ≤ j < s -1, n+1-s n-j if s -1 ≤ j ≤ n.
By face number of a tile, we mean its number of open simplices of the corresponding dimension.

Proof. We proceed by induction on the dimension n. If n = 1, one checks the result. Now let us suppose that the result holds true for n ≥ 1. By definition, T n+1 0 = ∆ n+1 , so that for every j ∈ {0, . . . , n + 1}, f j (T n+1 0 ) = n+2 j+1 = n+2 n+1-j . Hence the result for s = 0. Similarly,

T n+1 n+2 = • ∆n+1, so that f j (T n+1 n+2 ) = δ n+1j . Now, if 1 ≤ s ≤ n + 1, we know from Proposition 4.1 that T n+1 s = (c * T n s ) \ {c}.
The open faces of T n+1 s are thus either the faces of the basis T n s of the cone, or the cones over the faces of T n s . We deduce from Pascal's formula that for every j ∈ {0, . . . , n + 1},

f j (T n+1 s ) = f j (T n s ) + f j-1 (T n s ) = n+1-s n-j + n+1-s n+1-j = n+2-s n+1-j .

Tilings

For every n > 0, let T (n) be the set of n-dimensional simplicial complexes that can be tiled by T n 0 , . . . , T n n+1 .

Proposition 4.4.

1. Every pure finite simplicial complex of dimension one is tileable.

2. For every n > 0, if K ∈ T (n), then for every i ∈ {1, . . . , n}, the i-skeleton K (i) belongs to T (i). Moreover, any tiling of K induces a tiling on K (i) .

Recall that an n-dimensional simplicial complex is called pure if each of its simplex is a face of an n-simplex. The second part of Proposition 4.4 is the first part of Theorem 1.9.

Proof. For the first part we proceed by induction on the number of edges. If such a complex K contains only one edge, it is tiled by T 1 0 = ∆ 1 , since it is pure. Now let us suppose that the result holds true for every pure complex containing N edges. Let K be a pure simplicial complex of dimension one with N + 1 edges and let e be an edge of K. Thus, K = K ∪ e where K is a pure simplicial complex covered by N edges and by the hypothesis it can be tiled. We choose a tiling of K . Then, e has 0,1 or 2 common vertices with K . We then extend the tiling of K to a tiling of K by adding T 1 0 , T 1 1 or T 1 2 , respectively. And for the second part we proceed by induction on the dimension n. If n = 1 the result follows from the first part. Let us suppose that the result holds true for the dimension n. Let K be a tiled finite simplicial complex of dimension n + 1. The n-skeleton of K can be obtained by removing all open (n + 1)-simplices. These are exactly the interiors of the tiles of K. However, from Proposition 4.1, for every s ∈ {0, . . . , n + 2}, T n+1 s \ T n+1 n+2 is tiled by the tiles of dimension n. Thus, the n-skeleton of K gets an induced tiling of dimension n. The result follows from the fact that

K (i) = (K (n) ) (i) .
Proposition 4.4 raises the following question. Let X be a triangulated manifold with boundary. If X belongs to T (n), does ∂X belong to T (n -1)? Example 4.5.

1. Figure 2 shows some examples of one-dimensional tiled simplicial complexes.

Figure 2: Tiled one-dimensional simplicial complexes.

2. In dimension 2, ∂∆ 2 × [0, 1] can be tiled using six T 2 1 , by gluing three copies of the tiling shown in Figure 3. 3. The prism ∂(∆ 2 × [0, 1]) can be tiled using six T 2 1 and two T 2 3 , by gluing on the two boundary components of the previous example the open simplices T 2 3 .

4. The prism ∂(∆ 2 × [0, 1]) can likewise be tiled by six T 2 2 and two T 2 0 .

5. The cylindrical parts in examples 3 and 4 above can be glued together to produce a tiled two-torus.

Definition 4.6. A tiling of an n-dimensional simplicial complex is called regular if and only if it uses one tile T n 0 for each connected component.

Recall that a simplicial complex is called shellable if its maximal simplices can be arranged in linear order σ 1 , σ 2 , . . . , σ t in such a way that the subcomplex (∪ k-1 i=1 σ i ) ∩ σ k is pure and (dim σ k -1)-dimensional for all k = 2, . . . , t, see [START_REF] Kozlov | Combinatorial algebraic topology[END_REF] for instance. Proposition 4.7. Let K be a tiled finite n-dimensional simplicial complex that has a filtration

K 1 ⊂ K 2 ⊂ . . . ⊂ K N -1 ⊂ K N = K of
tiled subcomplexes such that for every i ∈ {1, . . . N }, K i contains i tiles. Then, the tiling of K is regular and its connected components are shellable and homotopy equivalent to n-dimensional spheres or balls.

(Shellable simplicial complexes are known to be homotopy equivalent to wedges of spheres, see Theorem 12.3 of [START_REF] Kozlov | Combinatorial algebraic topology[END_REF] for instance).

Proof. We proceed by induction on N . If N = 1, K = T n 0 = ∆ n , so that K has a regular tiling and is homeomorphic to a ball. Let us suppose that the result holds true for N and that K has a filtration

K 1 ⊂ . . . ⊂ K N ⊂ K N +1 = K. Then, K N +1 = K N ∪ T where T ∈ {T n 0 , . . . , T n n+1 }
, and K N is regular by the hypothesis. If K N and K N +1 do not have the same number of connected components, then T is a connected component of K so that T = T n 0 as in the case N = 1 and K N +1 is regular. Moreover, K N being homotopy equivalent to a union of n-dimensional spheres or balls, so is K N +1 , with one more ball component. Otherwise, K N and K N +1 have the same number of connected components. In this case, T is glued to one of the connected components of K N and thus T = T n 0 . Hence, K N +1 is regular, which proves the first part. Moreover, if T = T n n+1 =

• ∆, then since K N is the union of spheres or balls so is K N +1 , a homotopy ball of K N becoming a homotopy sphere. If T = T n n+1 , the boundary of T is not empty so that T intersects the boundary of K N +1 . In this case, K N is a deformation retract of K N +1 and is homotopy equivalent to K N +1 . Hence the result. Definition 4.8. Let K be an n-dimensional finite simplicial complex equipped with a tiling T . For every i ∈ {0, . . . , n + 1}, let h i (T ) be the number of tiles T n i of T . The vector (h 0 (T ), . . . , h n+1 (T )) is called the h-vector of T and the polynomial h T (X) = n+1 i=0 h i (T )X i its h-polynomial.

The following Theorem 4.9 and Corollary 4.10 complete Theorem 1.8. Theorem 4.9. Let K be a tileable n-dimensional finite simplicial complex. For every tiling

T of K, its h-polynomial satisfies n+1 i=0 h i (T )X n+1-i = n+1 i=0 f i-1 (K)(X -1) n+1-i provided f -1 (K)
is chosen to be equal to h 0 (T ). In particular, two tilings T and T have the same h-polynomial if and only if h 0 (T ) = h 0 (T ).

Proof. Let T be a tiling of K. From Proposition 4.3,

n+1 j=0 f j-1 (K)T j = n+1 j=0 T j j s=0 n+1-s j-s h s (T ) = n+1 k=0 h k (T ) n+1 j=s n+1-s j-s T j = n+1 s=0 h s (T ) n+1-s j=0 n+1-s j T n+1-j = T n+1 n+1 s=0 h s (T )(1 + 1 T ) n+1-s ,
where the second line follows from Proposition 4.3 and the convention f -1 (K) = h 0 (K). By letting X = T +1 T or equivalently T = 1 X-1 , we get

n+1 s=0 h s (T )X n+1-s = n+1 j=0 f j-1 (K)(X -1) n+1-j .
Hence the result.

Corollary 4.10. Let K be a connected n-dimensional finite simplicial complex equipped with a regular tiling T . Then, the h-vector of T coincides with the h-vector of K.

Proof. Since K is connected and T regular, h 0 (T ) = 1. But the h-polynomial of K is by definition the polynomial satisfying the relation in Theorem 4.9 with f -1 = 1. Hence the result.

Remark 4.11. Corollary 4.10 provides a geometric interpretation for the h-vector of tiled simplicial complexes. It is similar to the known one for shellable complexes.

Corollary 4.12. Let T be a tiling of an n-dimensional finite simplicial complex K. Then χ(K) = h 0 (T ) + (-1) n h n+1 (T ).

Proof. From Theorem 4.9,

n+1 i=0 h i (T )X n+1-i = n+1 i=0 f i-1 (K)(X -1) n+1-i with f -1 = h 0 (T )
. By letting X = 0 we get h n+1 (T ) = (-1) n+1 h 0 (T ) + (-1) n χ(K) as χ(K) = n i=0 (-1) i f i (K). Hence the result.

The Euler characteristic of an even dimensional finite tiled simplicial complex is thus positive. Corollary 4.2 provides a tiling of spheres in any dimensions and Example 5 a tileable triangulation of a two-torus. Which three-manifolds possess tileable triangulations?

Barycentric subdivision

Recall that for every n > 0 and s ∈ {0, . . . , n + 1}, T n s = ∆ n \ (σ 1 ∪ . . . ∪ σ s ) where σ i denotes a facet of ∆ n . We set Sd(T n s ) = Sd(∆ n ) \ ∪ s i=1 Sd(σ i ). The remaining part of the paper rely on the following key result. Theorem 4.13. For every n > 0 and every s ∈ {0, . . . , n + 1}, Sd(T n s ) is tileable. Moreover, it can be tiled in such a way that only Sd(T n 0 ) (resp. Sd(T n n+1 )) contains the tile T n 0 (resp. T n n+1 ) in its tiling and it contains exactly one such tile.

Proof. We proceed by induction on the dimension n > 0. If n = 1, the tilings Sd(T

1 0 ) = T 1 0 T 1 1 , Sd(T 1 1 ) = 2T 1 1 and Sd(T 1 2 ) = T 1 1
T 1 2 are suitable, see Figure 4.

Sd(T 1 0 ) Sd(T 1 1 ) Sd(T 1 2 )
Figure 4: Tilings of subdivided one-dimensional tiles. Now, let us assume that the result holds true for r ≤ n and let us prove it for r = n + 1. From Corollary 4.2, ∂∆ n+1 has a tiling n+1 s=0 T n s . We equip Sd(∂∆ n+1 ) = n+1 s=0 Sd(T n s ) with the regular tiling given by the induction hypothesis. Then, Sd(∆ n+1 ) gets a partition by cones over the tiles of Sd(∂∆ n+1 ) centered at the barycenter of ∆ n+1 where all the cones except the one over T n 0 are deprived of their center. From Proposition 4.1, this partition induces a regular tiling of Sd(∆ n+1 ) = T n+1 0 . For every s ∈ {1, . . . , n + 2}, we equip Sd(T n+1 s ) = Sd(∆ n+1 ) \ s-1 j=0 Sd(T n j ) with the tiling induced by removing the bases of all the cones over the tiles s-1 j=0 T n j ⊂ Sd(∂∆ n+1 ). From Proposition 4.1, these cones deprived of their bases are tiles so that we get as well a tiling of Sd(T n+1 s ). Moreover, when s > 0, the cone over the unique tile T n 0 of the tiling Sd(∂∆ n+1 ) is deprived of its basis, so that the tiling we get does not contain T n+1 0 . Finally, by the induction hypothesis the tiling of Sd(∂∆ n+1 ) contains a unique tile T n n+1 which is contained in the tiling of Sd(T n n+1 ) ⊂ Sd(∂∆ n+1 ). Thus, the tiling of Sd(T n+1 s ) contains the tile T n+1 n+2 only when s = n + 2 and in this case it contains only one such tile, since from Proposition 4.1, T n+1 n+2 is the cone over T n n+1 deprived of its base and its center. Hence the result.

The proof of Theorem 4.13 provides a tiling of all the subdivided tiles Sd(T n s ), s ∈ {0, . . . , n + 1}, in any dimension n. Let H n be the (n + 2) × (n + 2) matrix whose (s + 1)-st row is the h-vector of the tiling of Sd(T n s ). It follows from the proof of Theorem 4.13 that the s-th row of H n+1 is obtained by adding the first s -1 rows of H n shifted by one step to the right to the n + 3 -s last rows of H n .

Example 4.14. The matrices H n for n ≤ 5 are the following.

H 1 =   1 1 0 0 2 0 0 1 1   , H 2 =     1 4 1 0 0 4 2 0 0 2 4 0 0 1 4 1     , H 3 =      
        , H 5 =          
          .
We thus set H 0 = 1 0 0 1 , but do not consider the case n = 0 throughout the paper.

Corollary 4.15. Every tiling T (resp. every regular tiling) of a finite simplicial complex K induces a tiling Sd(T ) (resp. a regular tiling) on Sd(K). Moreover, h(Sd(T )) = h(T ) t H n .

Let Λ n = [λ ij ] be the lower triangular (n + 2) × (n + 2) matrix where λ ij denotes the number of j -2 interior faces of Sd(∆ i-2 ), 1 ≤ i, j ≤ n + 2. We agree that the standard simplex of dimension -1 has a unique face in dimension -1. The diagonal entries of Λ n are (i -1)!, i ∈ {1, . . . , n + 2}, see [START_REF] Brenti | f -vectors of barycentric subdivisions[END_REF][START_REF] Delucchi | Face vectors of subdivided simplicial complexes[END_REF][START_REF] Salepci | Asymptotic measures and links in simplicial complexes[END_REF]. Let F n be the (n + 2) × (n + 2) matrix whose s-th row is the face vector (f -1 (T n s-1 ), . . . , f n (T n s-1 )) of T n s-1 , where we set f -1 (T n s ) = 0 if s = 0 and f -1 (T n 0 ) = 1.

Theorem 4.16. For every n > 0, F n is unipotent upper triangular and

H n F n = F n Λ n . Moreover, H n = [h ij ] 1≤i,j≤n+2 satisfies the symmetry property h ij = h n+3-i,n+3-j .
Proof. It follows from Proposition 4.3 that F n = [ n+2-i n+2-j ] 1≤i,j≤n+2 , thus it is a unipotent upper triangular matrix. Now, by definition of H n and F n , the s-th row of the product H n F n is the face vector of Sd(T n s-1 ), s ∈ {1, . . . , n + 2}. Likewise, by definition of F n and Λ n , the s-th row of the product F n Λ n is the face vector of Sd(T n s-1 ), s ∈ {1, . . . , n + 2}. Thus, the two products coincide.

Finally, to prove the symmetry property of H n we proceed by induction. The matrix H 1 satisfies this symmetry. Let us now assume that H r satisfies the symmetry property for every r ≤ n. By definition, the s-th row of H n+1 is obtained by adding up the first s -1 rows of H n shifted once to the right with its n+3-s last rows. We deduce that for every 1 ≤ s, j ≤ n+3,

h n+1 sj = s-1 i=1 h n ij-1 + n+2 i=s h n ij = s-1 i=1 h n n+3-i,n+4-j + n+2 i=s h n n+3-i,n+3-j
, since H n satisfies the symmetry property by induction hypothesis. Thus,

h n+1 sj = n+2 i=n+4-s h n i,n+4-j + n+3-s i=1 h n i,n+3-j = h n+1 n+4-s,n+4-j .
Hence the result.

Let ρ n be the involution (h 0 , . . . , h n+1 ) ∈ R n+2 → (h n+1 , . . . , h 0 ) ∈ R n+2 . The symmetry property given by Theorem 4.16 means that the endomorphism H n of R n+2 commutes with ρ n . Corollary 4.17. For every n > 0, H n is diagonalizable with eigenvalues s!, s ∈ {0, 1, . . . , n + 1}. Moreover, the restriction of ρ n to the eigenspace of s! is (-1) n+1-s id. The vector (1, 1, . . . , 1) = (f n (T n 0 ), . . . , f n (T n n+1 )) spans the eigenspace of H n associated to the eigenvalue (n + 1)!.

Proof. By Theorem 4.16, H n is conjugated to the matrix Λ n which is diagonalizable with eigenvalues s!, s ∈ {0, . . . , n + 1}, see [START_REF] Brenti | f -vectors of barycentric subdivisions[END_REF][START_REF] Delucchi | Face vectors of subdivided simplicial complexes[END_REF][START_REF] Salepci | Asymptotic measures and links in simplicial complexes[END_REF]. The first part follows. Again by Theorem 4.16 we know that the last column of F n is an eigenvector of H n associated to the eigenvalue (n + 1)!. It is the vector (f n (T n 0 ), . . . , f n (T n n+1 )) = (1, . . . , 1) which is preserved by ρ n . The result is thus proved for n = 1, since 1 is an eigenvalue of multiplicity two and the signature of ρ n vanishes on the corresponding eigenspace. We will prove by induction that if ρ n acts as -id on the eigenspace associated to the eigenvalue s! of H n , then ρ n+1 acts as +id on the eigenspace associated to the eigenvalue s! of H n+1 . The result then follows, as the signature of ρ n is either 0 or 1 depending on the parity of n. Let s ≤ n be such that ρ n acts as -id on the eigenspace associated to the eigenvalue s! of H n . By Theorem 4.16, the eigenvectors associated to this eigenvalue are linear combination of the n+2-s last columns of F n . Indeed, let X n,s be an eigenvector of Λ n associated to the eigenvalue s!, then F n X n,s is an eigenvector of H n corresponding to s!. By hypothesis, ρ n acts as -id on the eigenspace spanned by this vector, so that (1, . . . , 1) t F n X n,s = 0. By definition of F n and from Corollary 4.2, (1, . . . , 1) t F n is the face vector f n of ∂∆ n+1 . As Λ n is lower triangular, there exists

x n+1 ∈ R such that X n+1,s = X n,s x n+1
. The first row of F n+1 is the face vector of ∆ n+1 = T n+1 0 mial q ∞ n = n p=0 q p,n T p , see [START_REF] Brenti | f -vectors of barycentric subdivisions[END_REF][START_REF] Delucchi | Face vectors of subdivided simplicial complexes[END_REF][START_REF] Salepci | Asymptotic measures and links in simplicial complexes[END_REF]. The first part of Corollary 4.18 is nothing but this result expressed in terms of h-vector and its proof is similar to the one of [START_REF] Delucchi | Face vectors of subdivided simplicial complexes[END_REF].

Corollary 4.19. For every n > 0, h n (X) = (X -1) n q n 1 X-1 , where h n (X) = n i=0 h n i X i . Proof. Let K be a finite n-dimensional simplicial complex equipped with a tiling T . For example, K = ∆ n being tiled with a single T n 0 . From Theorem 4.9 we know that |h(T )| = f n (K) and deduce

1 |h(T )|(n + 1)! d n+1 i=0 h i (Sd d (T ))X n+1-i = 1 f n (K)(n + 1)! d n+1 i=0 f i-1 (Sd d (K))(X -1) n+1-i with f -1 (Sd d (K)) = h 0 (Sd d (T ))
. From Theorem 4.13, h 0 (Sd d (T )) = h 0 (T ). We thus deduce from Corollary 4.18 and [4] by passing to the limit as d grows to +∞ that n+1

i=0 h n i X n+1-i = n+1 i=1 q i-1,n (X -1) n+1-i = (X -1) n q n ( 1 X-1
). The result now follows from Corollary 4.18, since h n 0 = h n n+1 = 0 and ρ n (h n ) = h n .

Packings

Theorem 5.1. For every n > 0, Sd(∆ n ) contains a packing of disjoint simplices with one n-simplex and for every j ∈ {0, . . . , n -1}, 2 n-1-j j-simplices. Moreover, for every s ∈ {1, . . . , n + 1}, Sd(T n s ) contains a packing of disjoint simplices with one (n + 1 -s)-simplex and if s ≤ n -1, for every j ∈ {0, . . . , n -1 -s}, 2 n-1-s-j simplices of dimension j. The simplex of dimension n + 1 -s reads [σ s-1 , . . . , σm ], where for i ∈ {s -1, . . . , n}, σ i denotes an i-simplex of ∆ n .

Proof. We proceed by induction on the dimension n > 0. If n = 1, one checks the result, Sd(∆ 1 ) ⊃ ∆ 1 ∆ 0 , Sd(T 1 1 ) ⊃ ∆ 1 and Sd(T 1 2 ) ⊃ ∆ 0 . Suppose now that the result holds true for every dimension ≤ n. From Corollary 4.2, we know that ∂∆ n+1 is tileable and ∂∆ n+1 = n+1 s=0 T n s , so that Sd(∂∆ n+1 ) = n+1 s=0 Sd(T n s ). The union of the packings given by the induction hypothesis provides a packing of the boundary of Sd(∆ n+1 ) which contains two n-simplices. We replace the n-simplex contained in Sd(T n 0 ) by its cone centered at the barycenter of ∆ n+1 . We get in this way a packing of disjoint simplices in Sd(∆ n+1 ) containing a simplex of dimension n + 1, a simplex of dimension n and 1 + n-1-j s=0 2 n-1-s-j simplices of dimension j, j ∈ {0, . . . , n -1}. Now, 1 + n-1-j s=0 2 n-1-s-j = 1 + n-1-j s=0 2 s = 2 n-j . Likewise, from Proposition 4.1 we deduce that for every s ∈ {1, . . . , n + 1}, Sd(∂T n+1 s ) = n+1 l=s Sd(T n l ). The union of the packings given by the induction hypothesis provides a packing of simplices in Sd(T n+1 s )∩Sd(∂∆ n+1 ) which contains one simplex of dimension n+1-s, of the form [σ s-1 . . . , σn ], that we replace by its cone centered at the barycenter of ∆ n+1 . We thus get a packing of disjoint simplices in Sd(T n+1 s ) which consists of one simplex of dimension n + 2 -s, of the form [σ s-1 , . . . , σn+1 ], and one simplex of dimension n -s if s < n + 1 together with, if s ≤ n -1, 1 + n-1-s-j l=0 2 n-1-s-j-l = 2 n-s-j simplices of dimension j for every j ∈ {0, . . . , n -1 -s}. Finally, Sd(T n+1 n+2 ) contains the barycenter of ∆ n+1 , which can be written as σn+1 . Hence the result.

We now able to prove Theorem 1.12.

Proof of Theorem 1.12. By Corollary 4.15, Sd(K) is tiled by Sd(T ) so that the union of the packings given by Theorem 5.1 provides the result.

We may relax the condition to be disjoint in Theorems 5.1 and 1.12, to get the following results.

Theorem 5.2. For every n > 0, every p ∈ {1, . . . , n-1} and every s ∈ {0, . . . , n+1}, Sd(T n s ) contains a packing of simplices with one simplex of dimension n + 1 -s + p if s ≥ p + 1 or 2 p-s simplices of dimension n if s ≤ p together with 2 n-1-s-j+p simplices of dimension j, j ∈ {p, . . . , min(n -1 -s + p, n -1)} if s ≤ n -1, in such a way that the intersection of two simplices of this collection is of dimension less than p and the intersection of each simplex with Sd(∂∆ n ) \ Sd(T n s ) is of dimension less than p. Moreover, the (n + 1 -s + p)-simplex is of the form [σ s-p-1 , . . . , σn ], where for every i ∈ {s -p -1, . . . , n}, σ i is an i-simplex of ∆ n .

Proof. We proceed by induction on p ∈ {1, . . . , n}. If p = 1, we first check the result for n = 1. In this case, Sd(∆ 1 ) contains exactly two simplices of dimension one intersecting each other at the barycenter of ∆ 1 and intersecting Sd(∂∆ 1 ) at a vertex. This provides a suitable packing for Sd(T 1 0 ), Sd(T ). The union for l ∈ {s, . . . , n} of the packings given by Theorem 5.1 provides a packing of disjoint simplices in Sd(T n s ) ∩ Sd(∂∆ n ) which contains an (n -s)-simplex of the form [σ s-1 , . . . , σn-1 ], a simplex of dimension n -1 -s if s ≤ n -1 and if s < n -1, 2 n-1-s-j simplices of dimension j for every j ∈ {0, . . . , n -2 -s}. By replacing these simplices by their cones centered at the barycenter of ∆ n , we get a collection of simplices in Sd(T n s ) which contains one (n + 1 -s)-simplex of the form [σ s-1 , . . . , σn ] and if s ≤ n -1, 2 n-s-j simplices of dimension j for every j ∈ {1, . . . , n -s}. Moreover, two simplices of this collection intersect at the barycenter of ∆ n and these simplices are contained in Sd(T n s ). If s ≥ 2, we remark that an (n + 1 -s)-simplex is of the form [σ s-1 , . . . , σn ], where σ i is an i-simplex of ∆ n . By choosing a facet σ s-2 of σ s-1 and by replacing this (n + 1 -s)-simplex by the (n + 2 -s)-simplex [σ s-2 , . . . , σn ] we get the required packing. This last simplex indeed intersects Sd(∂∆ n ) \ Sd(T n s ) at the vertex {σ s-2 }. If s = 0, we likewise know from Proposition 4.1 that Sd(∂∆ n ) = n l=0 Sd(T n-1 l

). The union for l ∈ {0, . . . , n} of the packings given by Theorem 5.1 provides a packing of disjoint simplices in Sd(∂∆ n ) which contains two simplices of dimension n -1 and for every j ∈ {0, . . . , n -2}, 2 n-1-j simplices of dimension j. By replacing these simplices by their cones centered at the barycenter of ∆ n , we get a collection of simplices of Sd(T n 0 ) that consists of two simplices of dimension n and of 2 n-j simplices of dimension j, j ∈ {1, . . . , n -1}. Moreover, two such simplices intersect at the barycenter of ∆ n . Finally, if s = n + 1, the 1-simplex [σ n-1 , σn ], where σ i ⊂ ∆ n is of dimension i, intersects Sd(∂∆ n ) at the vertex {σ s-1 }. This simplex gives the desired collection and the result follows for p = 1 and every n > 0. Now, let us suppose that the result holds true for every r ≤ p -1 and n ≥ p -1 and let us prove it for r = p and n ≥ p. ) with a packing given by the induction hypothesis applied to p-1 ≤ n-1. The union of these packings gives a packing of simplices of Sd(∂T n s ) such that the intersection between two simplices is of dimension ≤ p-1. This packing contains 2 n-2-s-j+p simplices of dimension j for every j ∈ {p -1, min(n -2 -s + p, n -2)} if s ≤ n -1 and one simplex of dimension n -2 -s + p, one simplex of dimension n -1 -s + p if s ≥ p and 1+ p-1 l=s 2 p-1-l simplices of dimension n-1 if s < p. By replacing all these simplices by their cones centered at the barycenter of ∆ n , we get a packing of simplices in Sd(T n s ) which contains 2 n-2-s-j+p simplices of dimension j +1 for every j ∈ {p-1, min(n-2-s+p, n-2)}, that is to say 2 n-1-s-j+p simplices of dimension j for j ∈ {p, min(n -1 -s + p, n -1)} if

3 .

 3 . , θq ] by deformation onto [ θ0 , . . . , θlm(θ) ]. For every face σ ∈ θ such that l m (σ) = -1, the restriction of r θ to σ is r σ . Proof. By definition, B d,m is the union of the open simplices σ of Sd d+1 (K) such that l m (σ) = -1. An open face of θ is thus included in B d,m if and only if it contains a vertex θi with i ≤ l m (θ). Hence the first part. The second and third parts follow from the definition of r θ .

  which is closed. Hence the first part. The third part of Proposition 2.4 guarantees that the retractions (r θ ) θ∈L glue together to define r L : L ∩ B d,m → L ∩ B d,m and the second part of Proposition 2.4 guarantees that the latter is a retraction of

5 .

 5 Dividing by (n + 1)! d , we deduce from Proposition 2.6 the upper bound

• ∆n and T n 0 Figure 1 :Corollary 4 . 2 .Proposition 4 . 3 .

 014243 Figure 1: The tiles in dimensions less than 4.

Figure 3 :

 3 Figure 3: A union of two T 2 1 .

  Let 2 ≤ p ≤ n and s ∈ {0, . . . , n}. From Proposition 4.1, Sd(∂T n s ) = n l=s Sd(T n-1 l ). Let us equip each Sd(T n-1 l

  Introduction 1.1 Average Betti numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Tilings and Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	2 Asymptotic behavior of the expected Betti numbers
	2.1 Proofs of Theorems 1.2 and 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . .
	2.2 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	2.3 Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	3 Refined upper bounds
	3.1 Induced filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	3.2 Monotony theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	4 Tilings
	4.1 Tiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	4.2 Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	4.3 Barycentric subdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	5 Packings

  for which there exists a path in V connecting ∂∆ n to ∂σ. In other words, ∈ P d,m (σ) if and only if V \ Theorem 2.7. Let ν ∈ [0, 1]. If there exist n > 0, k ∈ {1, . . . , n} and p ∈ {0, . . . , n -k} such that e p,ν (n, k) > ẽp,ν (n, k), then, n ≥ 3, p ≥ 1 and lim m→+∞ µ ν

	• σ percolates between
	the boundaries of ∆ n and σ.

  1 1 ) and Sd(T1 2 ). If n > 1, we know from Proposition 4.1 that for every s ∈ {1, . . . , n}, Sd(∂T n s ) =

	n l=s Sd(T n-1 l

Acknowledgement: The second author is partially supported by the ANR project MI-CROLOCAL (ANR-15CE40-0007-01).

and thus equals (f t n , 1). The first coefficient of F n+1 X n+1,s is thus f t n X n,s + x n+1 = x n+1 . However, the last row of F n+1 is the face vector of • ∆n= T n+1 n+2 so that the last coefficient of F n+1 X n+1,s is x n+1 as well. Similarly, the second row of F n+1 is the face vector of T n+1 1 . From Proposition 4.1, T n+1 0 = T n+1 1 ∪ T n 0 so that this second row differs from the first one by the face vector of T n 0 . Let us denote by F i n the rows of F n , i ∈ {1, . . . , n + 2}. We deduce that F 2 n+1 X n+1,s = F 1 n+1 X n+1,s -F 1 n X n,s = x n+1 -F 1 n X n,s . Since by Proposition 4.1 we have

Therefore, the second coefficient of the eigenvector F n+1 X n+1,s coincides with its second to last. Proceeding in the same way by induction, we deduce that the i-th coefficient of F n+1 X n+1,s coincides with the (n + 4 -i)-th one for i ∈ {1, . . . , n + 3}, which means ρ n+1 (F n+1 X n+1,s ) = F n+1 X n+1,s . Hence the result.

Let h n = (h n 0 , . . . , h n n+1 ) be the eigenvector of the transposed matrix H t n associated to the eigenvalue (n + 1)! and normalized in such a way that n+1 s=0 h n s = 1. For every tiled finite simplicial complex (K, T ) of dimension n, we set |h(T )| = n s=0 h s (T ). Corollary 4.18. For every tiled finite n-dimensional simplicial complex (K, T ), the sequence

) converges to h n as d grows to +∞. Moreover, h n 0 = h n n+1 = 0 and h n is preserved by the symmetry ρ n . Finally,

Proof. From Corollary 4.17 we know that H n = P DP -1 , where P denotes the matrix of eigenvectors of H n and D the diagonal matrix of eigenvalues s!, s ∈ {0, . . . , n + 1}, so that (P -1 ) t is a matrix of eigenvectors of H t n . Thus,

)! d converges to Diag(0, . . . , 0, 1) as d grows to +∞, compare [START_REF] Delucchi | Face vectors of subdivided simplicial complexes[END_REF]. Therefore, H d n (n+1)! d converges to the product (1, . . . , 1) t h, where h denotes an eigenvector of H t n associated to the eigenvalue (n + 1)!, since (1, . . . , 1) is an eigenvector of H n associated to the eigenvalue (n + 1)! from Corollary 4.17. For every d > 0, (1, . . . , 1) is preserved by H d n (n+1)! d , so that it is also preserved in the limit by (1, . . . , 1)h t . Thus, |h| = 1 and since the eigenvalue (n + 1)! is simple from Corollary 4.17, we deduce that h = h n . From Corollary 4.15 we know that for every d > 0, h(Sd

The fact that h n is preserved by ρ n follows from Corollary 4.17, since h n is an eigenvector of H t n associated to the eigenvalue (n + 1)! and the matrix J of ρ n is symmetric. Indeed, Corollary 4.17 implies that JP = P Diag((-1) n+2-i ), so that P -1 J = Diag((-1) n+2-i )P -1 and J t (P -1 ) t = (P -1 ) t Diag((-1) n+2-i ). As J t = J and the last column of (P -1 ) t is h n , ρ n fixes h n . (Another way to see that ρ n (h n ) = h n is to consider a basis (e 0 , . . . , e n+1 ) of eigenvectors of H n . Then the dual basis (e * 0 , . . . , e * n+1 ) is made of eigenvectors of H t n . If ρ n e i = i e i , then ρ t n e * i = i e * i , with i ∈ {±1} and ρ t n = ρ n under canonical identification between (R n+2 ) * and R n+2 .) Finally, by induction on d we deduce from Theorem 4.13 that the number of tiles T n 0 and T n n+1 which are in the tiling Sd(T ) does not depend on d so that h n 0 = lim d→+∞

Recall that for every finite n-dimensional simplicial complex K with face polynomial q K (T ) = n p=0 f p (K)T p , the polynomial

A packing of Sd(T n n+1 ) reduced to this simplex is suitable. Hence the result.

Corollary 5.3. Let K be a finite n-dimensional simplicial complex equipped with a tiling T of h-vector (h 0 (T ), . . . , h n+1 (T )) and let

simplices of dimension n in Sd(K) in such a way that they intersect each other in dimension less than p. Moreover, this packing can be completed by h n+1-j+p (T )+ 2 n-1-j+p n-1-j+p s=0 hs(T ) 2 s simplices of dimension j having the same property, j ∈ {p, . . . , n -1}.

Proof. By Corollary 4.15, Sd(K) is tiled by Sd(T ) so that the union of the packings given by Theorem 5.2 provides the result.

Remark 5.4.

1. Corollary 5.3 remains valid in the case p = 0 using the convention that simplices intersect in negative dimension when they are disjoint. The case p = 0 then gives back Theorem 1.12.

2. For every tiled finite n-dimensional simplicial complex (K, T ), e.g. K = ∆ n , and every 0 ≤ p ≤ n, Theorem 1.12 and Corollary 5.3 provide a sequence of packings in Sd d (K), d > 0. Corollary 4.18 provides the asymptotic of the number of simplices in each dimension of this sequence.

We finally prove Theorem 1.13.

Proof of Theorem simplices of dimension n together with h n+p-j (Sd d-1 (T )) + 2 n-2+p-j n-2+p-j i=0 h i (Sd d-1 (T )) 2 i simplices of dimension j, j ∈ {p -1, . . . , n -1}.

From Corollary 4.18, we know that for every i ∈ {0, . . . , n}, h i (Sd d-1 (T )) (n+1)! d converges to

h n i (n+1)! as d grows to +∞ and that h n 0 = h n n+1 = 0. For every p ∈ {1, . . . , n -1} and every j ∈ {p + 1, . . . , n}, M p,ν (∆ j ) = ν(1 -ν)(ν p + (1 -ν) p ) j p+1 by Lemma 3.5. The result follows after the change of variables j → n + p -j.


David Jondreville

A LOCALLY COMPACT QUANTUM GROUP ARISING FROM QUANTIZATION OF THE AFFINE GROUP OF A LOCAL FIELD

Keywords: Equivariant quantization, Locally compact quantum groups, Local fields

Using methods coming from non-formal equivariant quantization, we construct in this short note a unitary dual 2-cocycle on a discrete family of quotient groups of subgroups of the affine group of a local field (which is not of characteristic 2, nor an extension of Q 2 ). Using results of De Commer about Galois objects in operator algebras, we obtain new examples of locally compact quantum groups in the setting of von Neumann algebras.

Introduction

The construction of concrete examples of locally compact quantum groups (LCQG) in the setting of von Neumann algebras [START_REF] Mackey | Induced Representations of Locally Compact Groups I[END_REF][START_REF] Neshveyev | Deformation of C*-algebras by cocycles on locally compact quantum groups[END_REF] is still of upmost importance. The seminal work of De Commer [START_REF] Gayral | Quantization of the affine group of a local field[END_REF] provides a way to construct a new LCGQ from a Galois object on a given LCQG. An important class of Galois objects are associated with 2-cocycles [3, Section 5]. Recall that a unitary dual 2-cocycle on a LCQG (N , ∆), with dual quantum group ( N , ∆) (we use the standard notations, see e.g. [START_REF] Gayral | Quantization of the affine group of a local field[END_REF][START_REF] Vladimirov | P-adic analysis and mathematical physics[END_REF]), is a unitary element F ∈ N ⊗ N satisfying the cocycle relation: [START_REF] Bruhat | Distributions sur un groupe localement compact et applications à l'étude des représentations des groupes p-adiques[END_REF] (1 ⊗ F )(ι ⊗ ∆)(F ) = (F ⊗ 1)( ∆ ⊗ ι)(F ).

At the level of an ordinary locally compact group G, the explicit construction of a unitary dual 2-cocycle F is already a non-trivial task. In that case, F is a unitary element of the group von Neumann algebra W * (G × G) and the cocycle relation is equivalent (see [START_REF] Vladimirov | P-adic analysis and mathematical physics[END_REF]Section 4]) to the associativity of the following left-equivariant deformed product of the Fourier algebra A(G):

f 1 f 2 (g) = f1 ⊗ f2 λ g ⊗ λ g F * .
In the formula above, λ is the left regular representation λ g (f )(g ) = f (g -1 g ) and f → f is the standard identification of A(G) with the pre-dual of W * (G).

Hence, there is a simple strategy to construct dual 2-cocycles on groups. One starts with a left-equivariant and associative product on A(G) (or on a dense subalgebra of it), written in term of a distribution (in the sense of Bruhat [START_REF] Commer | Galois objects and cocycle twisting for locally compact quantum groups[END_REF]) K on G × G as follows:

f 1 f 2 = G×G K(g 1 , g 2 ) ρ g1 (f 1 ) ρ g2 (f 2 ) dg 1 dg 2 ,
where ρ is right regular action ρ g (f )(g ) = f (g g) and dg is a left-invariant Haar measure. One then considers the following (formal) convolution operator: All what remains to do then, is to verify that this operator makes sense and is unitary on L 2 (G) (for the left-invariant Haar measure). Now, the point is that an associative and left-equivariant product on a group G can be constructed out of a non-formal, G-equivariant quantization map on G. By this we mean a triple (Ω, H, α) consisting in a unitary operator Ω from L 2 (G) to the Hilbert space L 2 (H) of Hilbert-Schmidt operators on a separable Hilbert space H, together with an action of G on B(H) satisfying the covariance relation:

F := G×G K(g 1 , g 2 ) λ g -1 1 ⊗ λ g -1 2 dg 1 dg 2 .
Ω λ g (f ) = α g Ω(f ) , ∀f ∈ L 2 (G).
One can then define an associative and left-equivariant product on L 2 (G) by transporting the product of L 2 (H) to L 2 (G):

f 1 f 2 := Ω * Ω(f 1 ) Ω(f 2 ) .
In [START_REF] Kustermans | Locally compact quantum groups[END_REF], we have constructed such a non-formal, equivariant quantization map and the aim of this short note is to check that the associated candidate for a unitary dual 2-cocycle satisfies this unitary property. The family of groups we consider here are quotient groups of subgroups of the affine group of a local field. To simplify a little this introductive presentation, let us consider the case of the field of p-adic numbers Q p with p an odd prime number. Denoting by Z p the ring of integers of Q p , our groups are unimodular semidirect products of the form (1+p n Z p ) (Q p /p -n Z p ). We then have a unitary quantization map (see [START_REF] Kustermans | Locally compact quantum groups[END_REF]Proposition 3.9])

Ω : L 2 (1 + p n Z p ) (Q p /p -n Z p ) → L 2 L 2 (1 + p n Z p ) , defined for f ∈ L 1 ∩ L 2 (1 + p n Z p ) (Q p /p -n Z p ) by Ω(f ) = p n [t]∈Qp/p -n Zp 1+p n Zp f (a, [t]) π(a, t)Σπ * (a, t) da.
Here Σ is the (bounded) operator on L 2 (1 + p n Z p ) given by Σϕ(a) = ϕ(a -1 ), π is the irreducible unitary representation of (1 + p n Z p ) Q p (a subgroup of the affine group of Q p ) given by π(a, t)ϕ(a 0 ) := Ψ(a -1 0 at) ϕ a -1 a 0 , and Ψ is the basic unitary character of (Q p , +) given by

Ψ(x) = e 2iπ -1 n=-k anp n if x = ∞ n=-k a n p n ∈ Q p , a n ∈ {0, 1, . . . , p -1}.
The candidate for the unitary dual 2-cocycle associated with this quantization map is given by:

F := p 2n [t1],[t2] Ψ (a 1 -a -1 1 )t 2 -(a 2 -a -1 2 )t 1 λ (a1,[t1]) -1 ⊗ λ (a2,[t2]) -1 da 1 da 2 ,
where the sum is over (Q p /p -n Z p ) 2 and the integral is over (1 + p n Z p ) 2 . Our main result here, Theorem 3.4, is that F indeed defines a unitary operator, the 2-cocyclicity being automatic. However, for the sake of completeness, we give a direct proof of 2-cocyclicity in the Appendix. As mentioned earlier, De Commer's results then give an example of a locally compact quantum group in the von Neumann algebraic setting. Moreover, the results of [START_REF] Vladimirov | P-adic analysis and mathematical physics[END_REF] apply too and give a deformation theory for actions of that groups on C * -algebras.

Preliminaries

In this section, we first introduce the basic notations and tools we need. We then give a review of the pseudo-differential calculus constructed in [START_REF] Kustermans | Locally compact quantum groups[END_REF] and extract a dual 2-cocycle out of the associated -product (that is, the composition law of symbols). This pseudo-differential calculus is termed the p-adic Fuchs calculus because it imitates Unterberger's Fuchs calculus [14] in the non-Archimedean setting.

2.1 Setting and tools. Let k be a non-Archimedean local field. In characteristic zero, k is isomorphic to a finite degree extension of the field of p-adic numbers Q p . In positive characteristic, k is isomorphic to a field of Laurent series F ((X)) with coefficients in a finite field F . We let |.| k be the ultrametric absolute value, O k be the ring of integers of k (the unit ball), be a generator of its unique maximal ideal and O × k be the multiplicative group of units (the unit sphere). We let p be the characteristic of the residue field O k / O k and q = p f its cardinality. The uniformizer then satisfies | | k = q -1 . For every integer n ≥ 1, we denote by

U n := 1 + n O k ⊂ O ×
k the group of higher principal units. Note that U n is an open and compact subgroup of k × and it acts by dilations on k. We can therefore consider the semidirect product G n := U n k, that we view as a subgroup of the affine group k × k, with group law

(2) (x, t).(x , t ) := (xx , x -1 t + t ) , x, x ∈ k × , t, t ∈ k.
The group G n will be the covariance group of our pseudo-differential calculus. The subgroup U n will play the role of the configuration space and the role of the phase space will be played by the homogeneous space

X n := G n /H n , where H n is the closed subgroup {1} × -n O k ⊂ G n . Since H n is normal in G n ,
the quotient X n is endowed with a group structure. This is on the group X n that we will construct a unitary dual 2-cocycle. Of course, X n is isomorphic to the semidirect product U n Γ n , where Γ n := k/ -n O k is discrete and Abelian. As a locally compact group, our phase space X n is not compact, nor discrete and non-Abelian. We will denote the elements of X n by pairs (a, [t]), where a ∈ U n and [t] := t + -n O k ∈ Γ n , so that the group law of X n is given by (a, [t]).(a , [t ]) := (aa , [a -1 t + t ]).

To simplify the notations, we will frequently denote an element of G n by g := (a, t) and an element of the quotient group X n by [g] := (a, [t]). We normalize the Haar measure dt of k so that Vol(O k ) = 1. Since U n is open in k × , the Haar measure of U n is given by the restriction of the Haar measure of k

× (which is |t| -1 k dt) but since U n ⊂ O × k , it is just dt. The G n -invariant measure of X n (
which is also a Haar measure for the quotient group structure), denoted by d[g], is chosen to be the product of the Haar measure of U n by the counting measure on Γ n . In particular, both groups G n and X n are unimodular. Once for all, we fix a unitary character Ψ of the additive group k which is required to be trivial on O k but not on -1 O k . We also deduce by [START_REF] Kustermans | Locally compact quantum groups in the von Neumann algebraic setting[END_REF]Theorem 23.25] that Γ n is isomorphic to the additive group n O k .

Important ingredients of the p-adic Fuchs calculus, are the square function

σ : U n → U n , a → a 2 ,
and the hyperbolic sine type function

φ : U n → n O k , a → a -a -1 .
The crucial requirement for the p-adic Fuchs calculus to behave well, is that the mappings σ and φ are C 1 -homeomorphisms (in the precise sense of [12, p.77]). For this property to hold true, we need to assume that the element 2 ∈ k belongs to O × k . Equivalently ([5, Proposition 2.1]), we need to exclude two cases: when k is of characteristic two and when k is an extension of Q 2 . Then, σ and φ become surjective isometries of U n viewed as metric space for the induced ultra-metric of k.

We then define the square root function

U n → U n , a → a 1 2
, as the reciprocal mapping of the square function σ :

U n → U n . It is then shown in [5, Proposition 2.1] that |σ | k = |φ | k = 1.
Combining [12, p.287] (which only treats the one-dimensional case) and [13, p.49] (which only treats the case of Q d p ), we deduce the following substitution formula:

V f (t) dt = U f • ϕ(t) |Jac ϕ (t)| k dt, where U, V are compact open subsets of k d , f : V → C is L 1 and ϕ : U → V is a C 1 -homeomorphism such that Jac ϕ (t) = 0 for all t ∈ U .
In particular, we will frequently use this formula for ϕ = σ and ϕ = φ, which in these cases gives:

(3)

Un f (σ(a)) da = Un f (a) da and Un h(φ(a)) da = n O k h(x) dx.
With our choice of normalization for the Haar measure of k and with the non-trivial previously fixed standard character Ψ (that is constant on O k but not on -1 O k ), the Fourier transform

F k f (x) := k f (t) Ψ(xt) dt, f ∈ L 1 (k) ∩ L 2 (k), extends to a unitary operator on L 2 (k). Identifying a function f ∈ L 1 (Γ n ) with the function f ∈ L 1 (k) invariant under translations in -n O k , we therefore have (4) [t]∈Γn f ([t]) = q -n k f (t) dt. In particular, if f ∈ L 1 (k) is invariant by translations in -n O k then F k (f ) is supported on n O k and using the identification Γ n ∼ n O k , we get F k f = q n F Γn f , (5)
where

F Γn : L 2 (Γ n ) → L 2 ( n O k ), f → x → [t]∈Γn f ([t]) Ψ(xt) ,
is the Fourier transform on Γ n . We denote by D(k) the space of Bruhat-test functions on k [START_REF] Commer | Galois objects and cocycle twisting for locally compact quantum groups[END_REF]. Since k is totally disconnected, D(k) coincides with the space of locally constant compactly supported functions on k. Recall also that the Fourier transform F k is a homeomorphism of D(k). Similarly, we let D(X n ) be the space of Bruhat-test functions X n , that is the space of functions on X n which are locally constant in the continuous variable a ∈ U n and have finite support in the discrete variable [t] ∈ Γ n .

2.2

The p-adic Fuchs calculus. With Ψ the fixed unitary character of (k, +), we define the following representation π of the covariance group G n on the Hilbert space L 2 (U n ) of square integrable functions on the configuration space: [START_REF] Kustermans | Locally compact quantum groups in the von Neumann algebraic setting[END_REF] π(a, t)ϕ(a 0 ) := Ψ(a -1 0 at) ϕ a -1 a 0 . From Mackey theory [START_REF] Schikhof | Ultrametric calculus[END_REF], any infinite dimensional unitary irreducible representation of G n is unitarily equivalent to a representation of this form (and they are classified by the orbits of the dual action of U n on k k). Let then Σ be the group-inversion operator on L 2 (U n ) (i.e. the antipode): Σ ϕ(a) := ϕ(a -1 ).

Note that Σ is bounded since U n is unimodular. (More precisely, Σ is unitary and self-adjoint, that is a symmetry.) Consider then the map

G n → B(L 2 (U n )), g → π(g) Σ π * (g).
Since Ψ| O k = 1, we observe that the above map is invariant under right-translations in H n . Therefore, it defines a map Ω :

X n = G n /H n → B(L 2 (U n )), explicitly given by Ω([g])ϕ(a 0 ) = Ψ φ(aa -1 0 )t ϕ(a 2 a -1 0
). The quantization map underlying the p-adic Fuchs calculus is then defined by

Ω : L 1 (X n ) → B(L 2 (U n )), f → q n Xn f ([g]) Ω([g]) d[g].
By construction, the quantization map is G n -covariant:

(7) π(g) Ω(f ) π * (g) = Ω(λ [g] f ).
Remark 2.1. Unterberger's Fuchs calculus is defined in a similar way, with the multiplicative group R * + replacing U n and with (the connected component of) the affine group R * + R replacing both G n and X n . The common point of the two situations, is that R * + and U n are the largest open subgroups of R * and of k × on which the square function is a homeomorphism.

2.3

The -product and the dual 2-cocycle. At this stage, the most important property of the quantization map (property requiring that σ and φ are homeomorphisms of class C 1 ), is that Ω extends (from L 1 (X n ) ∩ L 2 (X n )) to a surjective isometry from L 2 (X n ) to the Hilbert space of Hilbert-Schmidt operators on L 2 (U n ) (see [START_REF] Kustermans | Locally compact quantum groups[END_REF]Proposition 3.9]). We can therefore transport the associative algebra structure of Hilbert-Schmidt operators to L 2 (X n ):

: L 2 (X n ) × L 2 (X n ) → L 2 (X n ) , (f 1 , f 2 ) → Ω * Ω(f 1 ) Ω(f 2 ) .
It is proven in [START_REF] Kustermans | Locally compact quantum groups[END_REF]Proposition 3.13] that for

f 1 , f 2 ∈ L 2 (X n ) ∩ L 1 (X n ), the product f 1 f 2 ∈ L 2 (X n ) ∩ L ∞ (X n ) takes the form f 1 f 2 ([g 0 ]) = Xn×Xn K [g 0 ], [g 1 ], [g 2 ] f 1 ([g 1 ]) f 2 ([g 2 ]) d[g 1 ] d[g 2 ],
where K is the locally constant and bounded function on X 3 n given by:

K (a 1 , [t 1 ]), (a 2 , [t 2 ]), (a 3 , [t 3 ]) = q 2n Ψ φ(a 1 a -1 2 )t 3 + φ(a 2 a -1 3 )t 1 + φ(a 3 a -1 1 )t 2 .
By construction, the kernel K is invariant under the diagonal action of G n on X 3 n , thus invariant under the diagonal left action of X n on X 3 n . Hence (see [5, Proposition 3.14]) we have for

f 1 , f 2 ∈ L 2 (X n ) ∩ L 1 (X n
) and with ρ the right regular action:

f 1 f 2 = Xn×Xn K [e], [g 1 ], [g 2 ] ρ [g1] (f 1 ) ρ [g2] (f 2 ) d[g 1 ] d[g 2 ],
with

K [e], [g 1 ], [g 2 ] = q 2n Ψ -φ(a 1 )t 2 + φ(a 2 )t 1 . Note also that D(X n ) is stable under (see [5, Proposition 3.15]).
As explained in the introduction, the natural candidate for a unitary dual 2-cocycle F on the group X n , is given by the following convolution operator, initially defined as a quadratic form on D(X n × X n ):

F := q 2n Xn×Xn Ψ φ(a 1 )t 2 -φ(a 2 )t 1 λ [g1] -1 ⊗ λ [g2] -1 d[g 1 ] d[g 2 ]. ( 8
)
The goal of the next section is to prove that F preserves the Bruhat space D(X n × X n ) and extends to a unitary operator on L 2 (X n ). Once this will be proven, the associativity of on D(X n ) will immediately imply the 2-cocycle relation for F :

(1 ⊗ F ) Id ⊗ ∆(F ) = (F ⊗ 1) ∆(F ) ⊗ Id ,
where ∆ is the coproduct of the group von Neumann algebra W * (X n ), defined on the generators by ∆(λ

[g] ) = λ [g] ⊗ λ [g] .

Unitarity of the dual 2-cocycle

We start with an important factorization property:

Proposition 3.1. Let Ξ : U n × n O k × U n × n O k → U n × n O k × U n × n O k be the continuous map defined by: Ξ a 1 , x 1 , a 2 , x 2 := φ -1 (a -1 2 x 2 )a 1 , x 1 , φ -1 (-a -1 1 x 1 )a 2 , x 2 , and let T Ξ be the continuous operator on D(U n × n O k × U n × n O k )
given by:

T Ξ f = f • Ξ.
Then, the dual 2-cocycle F factorizes as:

F = Id ⊗ F -1 Γn ⊗ Id ⊗ F -1
Γn T Ξ Id ⊗ F Γn ⊗ Id ⊗ F Γn . Proof. Take f ∈ D(X n × X n ) and let [g j ] = (a j , [t j ]) ∈ X n , j = 1, 2. Then we have:

F f ([g 1 ], [g 2 ]) = q 2n Xn×Xn Ψ(-φ(a 2 )t 1 ) Ψ(φ(a 1 )t 2 ) f (a 1 , [t 1 ]).(a 1 , [t 1 ]); (a 2 , [t 2 ]).(a 2 , [t 2 ]) d[g 1 ]d[g 2 ] = q 2n Xn×Xn Ψ(-φ(a 2 )t 1 ) Ψ(φ(a 1 )t 2 ) f 1 a 1 a 1 , [a -1 1 t 1 + t 1 ]; a 2 a 2 , [a -1 2 t 2 + t 2 ] d[g 1 ]d[g 2 ].
The change of variables [t 1 ] → [a 1 (t 1 -t 1 )] and [t 2 ] → [a 2 (t 2 -t 2 )] leads to:

F f ([g 1 ], [g 2 ]) = q 2n Un×Un Ψ(φ(a 2 )a 1 t 1 ) Ψ(-φ(a 1 )a 2 t 2 ) × Id ⊗ F Γn ⊗ Id ⊗ F Γn f a 1 a 1 , -a 1 φ(a 2 ); a 2 a 2 , a 2 φ(a 1 ) da 1 da 2 .
The second formula in (3) then gives:

F f ([g 1 ], [g 2 ]) = q 2n n O k × n O k Ψ(x 1 t 1 ) Ψ(x 2 t 2 ) × Id ⊗ F Γn ⊗ Id ⊗ F Γn f φ -1 (a -1 2 x 2 )a 1 , x 1 ; φ -1 (-a -1 1 x 1 )a 2 , x 2 dx 1 dx 2 , which is the announced formula.
Since the operator of composition by a continuous function preserves the space of locally constant functions, we deduce: Corollary 3.2. The dual 2-cocycle F is continuous on the Bruhat space D(X n ×X n ). In particular, F also makes sense as a densely defined operator affiliated with the group von Neumann algebra

W * (X n × X n ).
Next, we come to our main technical result:

Lemma 3.3. The map Ξ defined in Proposition 3.1 is a homeomorphism of class C 1 with |Jac Ξ | k = 1.
Proof. The main difficulty is to show that Ξ is bijective. Due to the specific form of the map Ξ, it is equivalent to prove that for any (a

1 , x 1 , a 2 , x 2 ) ∈ U n × n O k × U n × n O k , there exists a unique (a 1 , a 2 ) ∈ U n × U n such that Ξ a 1 , x 1 , a 2 , x 2 = a 1 , x 1 , a 2 , x 2 . ( 9
)
The relation [START_REF] Schikhof | Ultrametric calculus[END_REF] give the (equivalent) systems of equations:

φ -1 (a -1 2 x 2 )a 1 = a 1 φ -1 (-a -1 1 x 1 )a 2 = a 2 ⇔ a -1 2 x 2 = φ(a 1 a -1 1 ) -a -1 1 x 1 = φ(a 2 a -1 2 ) ⇔ a -1 1 a 2 1 + a -1 2 x 2 a 1 -a 1 = 0 a -1 2 a 2 2 -a -1 1 x 1 a 2 -a 2 = 0 .
Note first that this system is invariant under the dilations:

(a 1 , a 2 , ; a 1 , a 2 , x 1 , x 2 ) → (α 1 a 1 , α 2 a 2 , ; α 1 a 1 , α 2 a 2 , α 1 x 1 , α 2 x 2 ), α 1 , α 2 ∈ U n .
Hence, we get the equivalent system of coupled quadratic equations in term of the homogeneous variables u j := a -1 j a j ∈ U n and parameters

X j := a -1 j x j ∈ n O k , j = 1, 2: (10) u 2 1 + X 2 u -1 2 u 1 -1 = 0 u 2 2 -X 1 u -1 1 u 2 -1 = 0 .
Note also that this system is invariant under the transformation

(u 1 , u 2 , X 1 , X 2 ) → (u 2 , u 1 , -X 2 , -X 1 ). ( 11
)
The discriminant of the first equation in [START_REF] Vladimirov | P-adic analysis and mathematical physics[END_REF] (treating u 2 ∈ U n as a parameter) is given by ∆

1 := 4 + X 2 2 u 2 2 = 4 1 + X 2 2 4u 2 2 . Since X 2 ∈ n O k , u 2 ∈ U n ⊂ O ×
k and since 4 is a unit too, we have:

1 + X 2 2 4u 2 2 ∈ 1 + 2n O k ⊂ 1 + n O k = U n ,
which therefore possesses a (unique) square root in U n . Since 4 possesses exactly two square roots in k (which are ±2), ∆ 1 also possesses exactly two square roots in k. Hence, the first equation of the quadratic system possesses exactly two solutions in k namely:

Y ± = ± 1 + X 2 2 4u 2 2 1 2 - X 2 2u 2 ∈ ±U n + n O k = ±1 + n O k .
By the symmetry (11) of the system, we also get:

u 1 = 1 + X 1 X 2 2 + X 2 2 2 -X 2 1 + X 1 2 + X 2 2 2 1 2 1 2 .
Finally, in term of the original variables and parameters, we get:

a 2 = a 2 1 + x 1 x 2 2a 1 a 2 + x 2 1 2a 2 1 + x 1 a 1 1 + x 1 2a 1 + x 2 2a 2 2 1 2 1 2 , a 1 = a 1 1 + x 1 x 2 2a 1 a 2 + x 2 2 2a 2 2 - x 2 a 2 1 + x 1 2a 1 + x 2 2a 2 2 1 2 1 2 .
This shows that Ξ is one-to-one and onto and the explicit expressions above entail that it is moreover a homeomorphism of class C 1 . It remains to compute the absolute value of the Jacobian of Ξ. Since for every a ∈ U n , φ (a) = 1 + a -2 , we have

(φ -1 ) (x) = 1 + φ -1 (x) -2 -1 , ∀x ∈ n O k .
As a consequence, we get Jac Ξ (a 1 , x 1 , a 2 , x 2 ) = φ -1 -a -1

1 x 1 φ -1 a -1 2 x 2 +a -1 1 a -1 2 1 + φ -1 -a -1 1 x 1 -2 -1 1 + φ -1 a -1 2 x 2 -2 -1 x 1 x 2 .
The first term clearly belongs to U n . For the second term, note first that

1 + φ -1 ± a -1 j x j -2 ∈ 1 + U n = 2 + n O k ⊂ O × k .
Hence in the second term, all the factors preceding x 1 x 2 are units. Hence the second term belongs to 2n O k . This proves that Jac Ξ takes values in U n + 2n O k = U n , from which the claim follows.

An immediate consequence of Lemma 3.3 is that the operator T Ξ is unitary on L 2 (U n × n O k × U n × n O k ). Since the partial Fourier transform Id⊗F Γn is unitary from L 2 (X n ) to L 2 (U n × n O k ), Proposition 3.1 immediately implies: Theorem 3.4. Let k be a local field which is not of characteristic 2, nor an extension of Q 2 . Then, the dual 2-cocycle F given in (8) is a unitary operator on L 2 (X n × X n ).

1

 1

Acknowledgments

This work is part of my PhD thesis. I would like to warmly thank my advisor Victor Gayral, for his extreme patience as well as his unshakeable support.

The solution Y -is not admissible because it does not belong to U n as it should be. Hence, we get only one solution, namely

Substituting this expression for u 1 in the second equation of the system (10) yields the relation:

which is equivalent to:

(13)

Squaring the relation (13) and multiplying it by u 2 2 give, in term of the variable U := u 2 2 ∈ U n , the following quadratic equation:

The discriminant of equation ( 14) is:

∆ 2 possesses exactly two square roots in k, given by

and thus, the equation ( 14) possesses exactly two solutions:

(15)

We need to prove that only one solution is admissible and this not that easy since here both U + and U -belong to U n . To this aim, note first that we can assume without loss of generality that X 1 = 0 (if X 1 = 0 there is no ambiguity). We then evaluate the relation (13) in U ± and multiply it by U ± to get:

we get, after simplification by X 1 = 0, the condition:

The first factor in the above expression belongs to U n while the second factor belongs to ±1+ n O k . This condition thus excludes U -. Hence, u 2 2 = U + , which only gives one solution since the square root function is bijective on U n :

A Proof of 2-cocyclicity

Our goal here is to provide a direct proof that the (unitary) convolution operator:

satisfies the 2-cocycle relation:

For convenience, we shall work with the adjoint of F , given on D(X n × X n ) by:

In term of F * , the cocycle relation reads:

Since f is compactly supported, the integrals above are, for fixed [g], [g ], [g ] ∈ X n , absolutely convergent. To simplify a little the notations, we shall let F Γn be the partial Fourier transform:

The dilation t 1 → -aa -1 1 t 1 followed by the translation t 1 → t 1 -t in the first expression and the dilation t 2 → -a a -1 2 t 2 followed by the translation t 2 → t 2 -t in the second, entail:

) . In term of these functions, we have

In order to compute Id ⊗ ∆(F * ) (1 ⊗ F * )f and ∆(F * ) ⊗ Id (F * ⊗ 1)f , we will substitute f by (1 ⊗ F * )f in (16) and f by (F * ⊗ 1)f in (17). To do that, we first need a convenient formula for F * h, with h ∈ D(X n × X n ). Using (partially) the factorization given in Proposition (3.1), we get:

4 , a 2 a -1 4 φ(a 3 ) da 3 da 4 . From this, we deduce:

Using that φ(a 3 )a -1 4 -φ(a 4 )a -1 3 = φ(a 3 a -1 4 ), ( 18) and ( 19) are respectively given by:

4 , a a -1 2 a -1 4 φ(a 3 ) da 3 da 4 , and by

2 , a a -1 2 φ(a 1 ) da 3 da 4 . By the dilation a 3 → a 4 a 3 and the change of variable x 3 = φ(a 3 ), ( 18) and (19) become:

2 , a a -1 2 φ(a 1 ) dx 3 da 4 . as an inverse Fourier transform, and we get:

Plugging these expressions in ( 16) and ( 17), we get after the cancellation of the Fourier transforms:

, aa 2 a -1 1 a -1 4 φ(a -1 4 ), a a -1 1 a -1 4 , a a -1 1 a -1 4 φ(a 4 a -1 2 ); a a -1 2 , a a -1 2 φ(a 1 ) da 1 da 2 da 4 . The last point is to show that the two above expressions are actually the same. Performing the dilations a 2 → a 4 a 2 and a 1 → a 2 a 1 , the last expression becomes:

1 φ(a 4 )t Ψ a a -1 1 a -1 2 a -1 4 φ(a 2 )t Ψ a a -1 2 a -1 4 φ(a 1 a 2 )t (F Γn ⊗ F Γn ⊗ F Γn )f aa -1 1 , -aa -1 1 φ(a 4 ), a a -1 1 a -1 2 a -1 4 , -a a -1 1 a -1 2 a -1 4 φ(a 2 ); a a -1 2 a -1 4 , a a -1 2 a -1 4 φ(a 1 a 2 ) da 1 da 2 da 4 , which, after the relabelling a 2 ↔ a 4 , is exactly the formula we found for Id ⊗ ∆(F * ) (1 ⊗ F * )f .