Leray Numbers of Tolerance Complexes

Let K be a simplicial complex on vertex set V. K is called d-Leray if the homology groups of any induced subcomplex of K are trivial in dimensions d and higher. K is called d-collapsible if it can be reduced to the void complex by sequentially removing a simplex of size at most d that is contained in a unique maximal face. Motivated by results of Eckhoff and of Montejano and Oliveros on “tolerant” versions of Helly’s theorem, we define the t-tolerance complex of K, Tt(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {T}}_{t}(K)$$\end{document}, as the simplicial complex on vertex set V whose simplices are formed as the union of a simplex in K and a set of size at most t. We prove that for any d and t there exists a positive integer h(t, d) such that, for every d-collapsible complex K, the t-tolerance complex Tt(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {T}}_t(K)$$\end{document} is h(t, d)-Leray. As an application, we present some new tolerant versions of the colorful Helly theorem.


Introduction
Let F be a finite family of non-empty sets.The nerve of F is the simplicial complex A simplicial complex K is called d-representable if it isomorphic to the nerve of a family of convex sets in R d .
Two closely related notions are the d-collapsibility and d-Lerayness of a complex, which were introduced by Wegner in [15] as generalizations of d-representability.Let V be a finite set, and let K be a simplicial complex on vertex set V .A face σ in K is said to be free if there exists exactly one maximal face of K that contains σ.Given a free face σ of cardinality at most d, we call the operation of removing from K all faces containing σ an elementary d-collapse.The complex K is called d-collapsible if all of its faces can be removed by performing a sequence of elementary d-collapses.The collapsibility number of K, denoted by C(K), is the minimum integer d such that K is d-collapsible.
For an integer i ≥ −1, let Hi (K) be the reduced i-dimensional homology group of K with coefficients in R. For U ⊂ V , the subcomplex of K induced by U is the complex K[U ] = {σ ∈ K : σ ⊂ U }.The complex K is called d-Leray if all its induced subcomplexes have trivial reduced homology groups in dimensions d and higher; that is, if Hi (K[U ]) = 0 for all i ≥ d and U ⊂ V .The Leray number of K, denoted by L(K), is the minimum integer d such that K is d-Leray.
It was shown in [15] that every d-representable complex is d-collapsible, and that every d-collapsible complex is d-Leray.See the survey paper by Tancer [13] for an overview of d-representability, d-collapsibility and d-Lerayness.
Helly's theorem [5] is a fundamental result in combinatorial geometry that asserts the following: for every finite family of convex sets in R d , if every subfamily of size at most d + 1 has a point in common, then the whole family has a point in common.See, for example, [1] for an overview of results and open problems related to Helly's theorem.
Let K be a complex on vertex set V .A missing face of K is a set τ ⊂ V such that τ / ∈ K but σ ∈ K for any σ τ .The Helly number of K, denoted by h(K), is the maximum dimension of a missing face of K. Helly's theorem is equivalent to the fact that, if K is d-representable, then h(K) ≤ d.
Since the boundary of a k-dimensional simplex has non-trivial homology in dimension k − 1, then every d-Leray complex K does not contain the boundary of a k-dimensional simplex as an induced subcomplex, for any k > d.This shows that the bound h(K) ≤ d also holds when K is d-Leray (and therefore the same holds when K is d-collapsible).
Let H be an r-uniform hypergraph on vertex set V .The covering number of H, denoted by τ (H), is the minimum size of a set U ⊂ V such that U intersects all the edges of H.The hypergraph H is called t-critical if τ (H) = t and τ (H ′ ) < t for every hypergraph H ′ that is obtained from H be removing an edge.The Erdős-Gallai number η(r, t) is the maximum number of vertices in an r-uniform t-critical hypergraph.Equivalently, η(r, t) is the minimum positive integer n such that for every r-uniform hypergraph H with τ (H) > t there exists an H ′ ⊂ H with |V (H ′ )| ≤ n such that τ (H ′ ) > t.Erdős and Gallai showed in [4] that η(2, t) = 2t and η(r, 2) = r+2 2 2 .For general r and t, Tuza proved in [14] the bound which is tight up to a constant factor.In particular, we have η(r, t) = O(t r−1 ) for r fixed and t → ∞, and η(r, t) = O(r t ) for t fixed and r → ∞.
Let F be a finite family of sets.We say that F has a point in common with tolerance t if there is a subfamily Montejano and Oliveros proved the following "tolerant version" of Helly's theorem.
Theorem 1.1 (Montejano and Oliveros [11,Theorem 3.1]).Let F be a finite family of convex sets in R d , and let t ≥ 0 be an integer.If every subfamily F ′ ⊂ F of size at most η(d + 1, t + 1) has a point in common with tolerance t, then F has a point in common with tolerance t.
In fact, it was shown in [11] that any family of sets satisfying a Helly property satisfies also a corresponding "tolerant Helly property".In terms of simplicial complexes, this may be stated as follows: Let K be a simplicial complex on vertex set V , and let t ≥ 0 be an integer.Define the simplicial complex We call T t (K) the t-tolerance complex of K.Note that T 0 (K) = K for every complex K.
It is natural to ask whether we can achieve a stronger conclusion by strengthening the assumptions on K.By replacing the Helly number with the collapsibility or Leray number, the following conjectures arise: Let t ≥ 1, and let A and B be two disjoint sets of size t + 1 each.Let K be the simplicial complex on vertex set A ∪ B whose maximal faces are the sets A and B. It is easy to check that K is 1-collapsible, and therefore 1-Leray (in fact, it is easy to show that it is even 1-representable).On the other hand, the complex T t (K) is the boundary of the simplex A ∪ B. That is, T t (K) is a 2t-dimensional sphere.In particular, it is not 2t-Leray.Therefore, for d = 1, the bound η(2, t + 1) − 1 = 2t + 1 in Conjectures 1.3 and 1.4 cannot be improved.
For t = 1, it was shown in [11,Theorem 3.2] that there exists a d-representable complex K such that T 1 (K) is the boundary of a In particular, T 1 (K) is not Our main result is the following: Theorem 1.5.Let K be a d-collapsible complex, and let t ≥ 0 be an integer.Then, T t (K) is h(t, d)-Leray, where h(0, d) = d for all d ≥ 0, and for t > 0, Note that we require the stronger property (collapsibility) for K, and obtain only the weaker property (Leray) for the tolerance complex.For d = 1, we obtain the tight bound h(t, 1) = 2t + 1 = η(2, t + 1) − 1.For d > 1, h(t, d) is larger than the conjectural bound η(d + 1, t + 1) − 1.However, when t is fixed, we have h(t, d) = O(d t+1 ), which is of the same order of magnitude as that of η(d + 1, t + 1) − 1.
In the special case d = 2, t = 1, we can prove the following stronger bound: Theorem 1.6.Let K be a 2-collapsible complex.Then, T 1 (K) is 5-Leray.

Organization
The paper is organized as follows.In Section 2 we present some basic facts about simplicial complexes, homology and collapsibility.In Section 3 we prove some auxiliary topological results that we will use later.In Section 4 we prove our main result, Theorem 1.5.In Section 5 we prove Theorem 1.6 about the Leray number of the 1-tolerance complex of a 2-collapsible complex.In Section 6 we present some applications to tolerant versions of the colorful Helly theorem.

Background
In this section we recall some basic definitions and results about simplicial complexes, homology and collapsibility.

Simplicial complexes
Let V be a finite set and let K ⊂ 2 V be a family of sets.K is called a simplicial complex For short, we call a k-dimensional simplex a k-simplex.Let K(k) be the set of all k-simplices.
The dimension of the complex K, denoted by dim(K), is the maximal dimension of a simplex in K.
K ′ is a subcomplex of K if it is a simplicial complex, and each simplex of K ′ is also a simplex of K.
Let U ⊂ V .The subcomplex of K induced by U is the complex Let τ ∈ K.We define the link of τ in K to be the subcomplex and the costar of τ in K to be the subcomplex Let X, Y be simplicial complexes on disjoint vertex sets.We define the join of X and Y to be the simplicial complex For U ⊂ V , we denote by 2 U = {σ : σ ⊂ U } the complete complex on vertex set U .

Simplicial homology
Let K be a simplicial complex.Let Hk (K) be the k-th reduced homology group of K with coefficients in R. We say that K is acyclic if Hk (K) = 0 for all k ≥ −1.
A useful tool for computing homology is the Mayer-Vietoris long exact sequence: Theorem 2.1 (Mayer-Vietoris).Let X, Y be simplicial complexes.Then, the following sequence is exact The following special case will be of use later: Theorem 2.2.Let K be a simplicial complex on vertex set V , and let v ∈ V .Then, the following sequence is exact . By Theorem 2.1, we have a long exact sequence Note that B is a cone over v, and therefore Hk (B) = 0 for all k.Moreover, A ∪ B = K and A ∩ B = lk(K, v).So, we obtain a long exact sequence Another useful way of computing homology is by the application of nerve theorems.Let X 1 , . . ., X m be simplicial complexes.The nerve of the family {X 1 , . . ., X m } is the simplicial complex where [m] := {1, 2, . . ., m}.Roughly speaking, given a family of simplicial complexes, a nerve theorem describes how much the topology of the nerve of the family reflects the topology of the union of the complexes, when every non-empty intersection of the complexes satisfies certain topological restrictions (see e.g.[3,Theorem 6] or [10, Theorem 2.1]).Here, we will use the following simple version of the nerve theorem: Theorem 2.3 (Leray's Nerve Theorem).Let X 1 , . . ., X m be simplicial complexes, and let for all k ≥ −1.
For the union of simplicial complexes, the Leray number can be bounded by the following result by Kalai and Meshulam.

Relative homology
Let X be a simplicial complex and let Y be a subcomplex of X.Let C k (X, Y ) be the R-vector space generated by the ordered k-simplices in X \ Y , under the relations for every k-simplex {v 0 , . . ., v k } ∈ X \ Y and permutation π : {0, . . ., k} → {0, . . ., k}.We define a linear map ) that acts on the spanning set by We define the group of k-cycles as Z k (X, Y ) = Ker(∂ k ) and the group of k-boundaries as We call H k (X, Y ) the k-th relative homology group of the pair Y ⊂ X.The relative homology of the pair Y ⊂ X is related to the homology of the two complexes via the following result: Theorem 2.5 (Long exact sequence of a pair).Let Y ⊂ X be simplicial complexes.Then, the following sequence is exact:
It will be convenient to use the following equivalent definition of d-collapsibility: Lemma 2.8 (Tancer [12, Lemma 5.2]).Let K be a simplicial complex.Then, K is dcollapsible if and only if one of the following holds: • There exists some σ ∈ K such that |σ| = d, σ is contained in a unique maximal face τ = σ of K, and cost(K, σ) is d-collapsible.

Some topological preliminaries
In this section we prove some auxiliary results on the homology groups of simplicial complexes that we will later need.
Using the Mayer-Vietoris exact sequence (Theorem 2.1) and Leray's Nerve Theorem (Theorem 2.3), we can prove the following.Lemma 3.1.Let X 1 , . . ., X m be simplicial complexes, and let X = m i=1 X i .If for all I ⊂ [m] of size at least 2, the complex ∩ i∈I X i is non-empty and acyclic, then for all k ≥ −1.
Proof.We argue by induction on m.For m = 1 the claim is trivial.Assume m > 1.Since i∈I X i is non-empty and acyclic for every I ⊂ [m − 1], we obtain, by the induction hypothesis, Hk we have by Theorem 2.1 a long exact sequence By the assumptions of this lemma, the nerve We will also need the following simple result about relative homology: Lemma 3.2.Let X be a simplicial complex on vertex set V , and let Y ⊂ X be a subcomplex.Assume that there is some σ ∈ X and subcomplexes Then, for all k.
Proof.For all k, let φ and extended linearly.Note that the maps φ k are linear isomorphisms.Denote by ∂ k the boundary operator of C k (X, Y ) and by ∂ ′ k the boundary operator of C k (Z, W ). We are left to show that φ is a chain map.That is, for any η ∈ Z(k) \ W (k), we need to show that ).Let η = {u 0 , . . ., u k }.For any i ∈ {0, . . ., k}, let η i = {u 0 , . . ., u i−1 , u i+1 , . . ., u k }.Then, since any subset of η ∪ σ belonging to X \ Y must contain σ, we have Hence, So C k (X, Y ) and C k−|σ| (Z, W ) are isomorphic as chain complexes, and in particular have isomorphic homology groups.
4 Proof of Theorem 1.5 In this section, we present the proof of Theorem 1.5.Note that the construction of the tolerance complexes depends on the vertex set of the original complex.For the construction of tolerance complexes, we will consider the vertex set of K[U ] to be the set U , the vertex set of cost(K, σ) to be V , and the vertex set of lk(K, σ) to be V \ σ.Lemma 4.1.Let K be a simplicial complex on vertex set V , and let σ ∈ K.Then, where Since σ ⊂ η 1 ∪(σ \σ ′ ) and |σ ′ ∪η 2 | ≤ t, we have τ ∈ T t (cost(K, σ)), which is a contradiction to the assumption τ ∈ T t (K) \ T t (cost(K, σ)).
By Lemma 3.2 and Lemma 4.1, we obtain: Corollary 4.2.Let K be a simplicial complex, and let σ ∈ K.Then, for all k, we have Proposition 4.3.Let K be a simplicial complex, and σ ∈ K such that σ is contained in a unique maximal simplex σ ∪ U ∈ K, where U = ∅.Then, for all k, By Corollary 4.2, we have By applying Theorem 2.5 to the pair T t (lk(K, σ)) ∩ Y ⊂ T t (lk(K, σ)), we obtain a long exact sequence In particular, since U = ∅, T t (lk(K, σ)) is contractible.Hence, We can write where In particular, it implies and hence, we conclude Since U = ∅, the intersection m i=1 Y W i is non-empty and acyclic.Therefore, by applying Lemma 3.1 to (4.1), we obtain Recall that h(t, d) is defined as follows: h(0, d) = d for all d ≥ 0, and for t > 0, Lemma 4.4.
Theorem 1.5.Let K be a d-collapsible complex, and let t ≥ 0 be an integer.Then, T t (K) is h(t, d)-Leray.
Proof.Let V be the vertex set of K. We will show that Hk (T t (K)) = 0 for k ≥ h(t, d).This is sufficient to prove the statement of the theorem, since ) and, by Lemma 2.6, K[W ] is d-collapsible for every W ⊂ V .We argue by induction on t.If t = 0 the statement obviously holds, since every d-collapsible complex is d-Leray.
Let t ≥ 1.We argue by induction on the size of K, that is, the number of simplices in K.If dim(K) < d, then dim(T t (K)) < d + t < h(t, d), so the statement holds.Otherwise, by Lemma 2.8, there is some σ ∈ K such that |σ| = d, σ is contained in a unique maximal face τ = σ of K, and cost(K, σ) is d-collapsible.
By Proposition 4.3, it is sufficient to show that, for every W ⊂ V \ (σ ∪ U ) of size t, the homology group Note that, for any σ ′ ⊂ σ, by Lemma 2.6 and Lemma 2.7, the complex lk(K, σ \ σ ′ )[U ∪ W ] is d-collapsible.Hence, by the induction hypothesis, for any σ In particular, 5 Improved bound for d = 2, t = 1 By Theorem 1.5 and Lemma 4.4, we proved that the 1-tolerance complex T 1 (K) is (d 2 +2d)-Leray for every d-collapsible complex K.This is of the same order of magnitude, but larger than the conjectural bound η In this section, we prove Theorem 1.6, which gives a tight bound for the Leray number of T 1 (K), in the special case that K is 2-collapsible.
For the proof we will need the following Lemma:

Tolerance in colorful Helly theorems
The colorful Helly theorem is one of the most important generalizations of Helly's theorem.It was observed by Lovász, and first appeared in Bárány's paper [2].It asserts the following.
Theorem 6.1 (Lovász, Bárány [2]).Let F 1 , F 2 , . . ., F d+1 be finite families of convex sets in R d , and let F be their disjoint union.Suppose every subfamily of F that contains exactly one member from each F i has a point in common.Then, some F i has a point in common.
Note that the colorful Helly theorem implies Helly's theorem, by assuming all F i 's are identical.In [11,Theorem 4.4], a tolerant version of the colorful Helly theorem in the plane was proved.Here is a more general statement.Theorem 6.2.Let F 1 , F 2 , . . ., F d+1 be finite families of convex sets in R d , and let F be their disjoint union.Suppose that every subfamily F ′ of F of size η(d + 1, t + 1) has a subfamily F ′′ ⊂ F ′ of size |F ′′ | ≥ |F ′ | − t such that any subfamily of F ′′ containing exactly one member from each F i has a point in common.Then, some F i has a point in common with tolerance t.
For completeness, we present a proof, closely following the argument presented in [11] for the special case d = 2, t = 1.
Proof of Theorem 6.2.Consider the hypergraph H whose vertex set is F and whose edges are the subfamilies that contain exactly one member from each F i and that do not have a point in common.
By the assumption of the theorem, every subhypergraph of H consisting of η(d+1, t+1) vertices has covering number at most t.Therefore, by the definition of η(d + 1, t + 1), H has covering number at most t.That is, there is a subfamily F ′ of F of size |F ′ | ≥ |F| − t such that every subfamily of F ′ consisting of exactly one set from each F i has a point in common.Therefore, by Theorem 6.1 (applied to the family F ′ ), there is some i such that F i ∩ F ′ has a point in common.Since |F i ∩ F ′ | ≥ |F i | − t, F i has a point in common with tolerance t.
Similarly as above, we observe that Theorem 6.2 implies Theorem 1.1, by assuming all F i 's are identical.
As an application of our main results, we obtain new tolerant variants of the colorful Helly theorem.
A family M of subsets of a non-empty set V is a matroid if it satisfies The rank function of a matroid M on V is a function ρ : 2 V → N such that for every W ⊂ V , ρ(W ) equals to the maximal size of W ′ ⊂ W with W ′ ∈ M .Note that the conditions (i) and (ii) allow us to regard a matroid M as a simplicial complex.The colorful Helly theorem can be generalized topologically as follows: Theorem 6.3 (Kalai and Meshulam, [6, Theorem 1.6]).Let K be a d-Leray complex on V and let M be a matroid on V with rank function ρ.If M ⊂ K, then there exists σ ∈ K such that ρ(V \ σ) ≤ d.
1, d) = d 2 + 2d.Indeed, this follows immediately from the definition of h(t, d) h(1, d) = d(h(0, d) + 1) + d = d 2 + 2d.Finally, we show that, for fixed t, h(t, d) = O(d t+1 ).We argue by induction on t.For t = 0 we have h(0, d) = d = O(d).Let t > 1.We will show that there is some constant C t such that, for large enough d, h(t, d) ≤ C t d t+1 .By the definition of h(t, d) and the induction hypothesis, we have,