Tverberg's theorem for cell complexes
Sho Hasui, Daisuke Kishimoto, Masahiro Takeda, Mitsunobu Tsutaya
aa r X i v : . [ m a t h . A T ] J a n TVERBERG’S THEOREM FOR CELL COMPLEXES
SHO HASUI, DAISUKE KISHIMOTO, MASAHIRO TAKEDA, AND MITSUNOBU TSUTAYA
Abstract.
The topological Tverberg theorem states that given any continuous map f : ∆ ( d +1)( r − → R d , there are pairwise disjoint faces σ , . . . , σ r of ∆ ( d +1)( r − such that f ( σ ) ∩ · · · ∩ f ( σ r ) = ∅ whenever r is a prime power. We generalize this theorem to acontinuous map from a certain CW complex into a Euclidean space. Introduction
Let d ≥ r ≥ d +1)( r − R d , there is a partition of points into r subsets whose convex hulls intersect.This theorem has been of great interest in combinatorics for more than 50 years. Clearly,Tverberg’s theorem can be restated in terms of an affine map from a ( d + 1)( r − R d . The topological Tverberg theorem replaces an affine map in Tverberg’s theoremby a continuous maps: for any continuous map f : ∆ ( d +1)( r − → R d , there are pairwisedisjoint faces σ , . . . , σ r of the simplex ∆ ( d +1)( r − such that f ( σ ) ∩ · · · ∩ f ( σ r ) = ∅ whenever r is a prime power. This was first proved by B´ar´any, Shlosman and Sz˝ucs [4]when r is a prime, and later by ¨Ozaydin [16] and Volovikov [17], independently, when r is aprime power. Remark that Frick [8] proved that the result does not hold unless we assume r is a prime power. See the surveys [3, 6] for more on the topological Tverberg theorem.In this paper we consider: Problem 1.1.
Can we replace a simplex in the topological Tverberg theorem by other CWcomplexes?There was a relevant problem posed by Tverberg [9]: can we replace a simplex by a polytopein the topological Tverberg theorem? This is affirmatively solved because the boundary ofany d -polytope is a refinement of the boundary of a d -simplex as proved by Gr¨unbaum [10,p. 200]. So this is not an essential generalization of the topological Tverberg theorem. Theproblem was also studied by B´ar´any, Kalai and Meshulam[2] and Blagojevi´c, Haase andZiegler [5] for matroid complexes. Mathematics Subject Classification.
Key words and phrases. topological Tverberg theorem, discretized configuration space, homotopy colimit.
We will prove a simplex in the topological Tverberg theorem can be replaced by the followingCW complex, where we will see the result is not a consequence of the topological Tverbergtheorem unlikely to the case of polytopes. To define such a CW complex, we set notationand terminology. A face of a CW complex will mean a closed cell. For pairwise disjoint faces σ , . . . , σ k of a CW complex X , let X ( σ , . . . , σ k ) denote a subcomplex of X consisting offaces disjoint from σ , . . . , σ k . A space Y is called n -acyclic if Y is non-empty and e H ∗ ( Y ) = 0for ∗ ≤ n , where n ≥
0. Clearly, n -connected spaces are n -acyclic. A non-empty space iscalled a ( − Definition 1.2.
We say that a regular CW complex X is r -complementary n -acyclic if X ( σ , . . . , σ k ) is non-empty and ( n − dim σ − · · · − dim σ k )-acyclic for each pairwise disjointfaces σ , . . . , σ k with dim σ + · · · + dim σ k ≤ n + 1, where k = 0 , , . . . , r .Examples of r -complementary n -acyclic complexes will be given in Section 2. Now we statethe main theorem. Theorem 1.3.
Let r be a prime power. If X is an ( r − -complementary ( d ( r − − -acyclic regular CW complex, then for any continuous map f : X → R d , there are pairwisedisjoint faces σ , . . . , σ r of X such that f ( σ ) ∩ · · · ∩ f ( σ r ) = ∅ . The topological Tverberg theorem for r = 2 is known as the topological Radon theorem.Clearly, we can replace a simplex by its boundary in the topological Tverberg theorem, soin the topological Radon theorem too. We will see in Corollary 2.3 below that every sim-plicial d -sphere is 1-complementary ( d − Corollary 1.4.
Let X be a simplicial d -sphere. Then for any continuous map f : X → R d there are disjoint faces σ , σ of X such that f ( σ ) ∩ f ( σ ) = ∅ . Gr¨unbaum and Sreedharan [11] gave a simplicial sphere which is not polytopal, the bound-ary of a polytope, and now most simplicial spheres in higher dimensions are known to benon-polytopal. Thus Corollary 1.4 does not follow from the topological Radon theorem, soTheorem 1.3 is a proper generalization of the topological Tverberg theorem.
Acknoledgement:
Hasui was supported by JSPS KAKENHI 18K13414, Kishimoto was sup-ported by JSPS KAKENHI 17K05248, and Tsutaya was supported by JSPS KAKENHI19K14535.
VERBERG’S THEOREM FOR CELL COMPLEXES 3 Examples
This section gives examples of r -complementary n -acyclic complexes. We start with asimplex: a d -simplex and its boundary are ( r − d − r )-acyclic. Then wesee that the topological Tverberg theorem is recovered by Theorem 1.3.Let X be a simplicial complex with vertex set V . For a non-empty subset I of V , the fullsubcomplex of X over I is defined to be the subcomplex of X consisting of all faces whosevertices are in I . By [15, Lemma 70.1], we have: Lemma 2.1.
Let X be a simplicial complex, and let X ( A ) denote the subcomplex of X consisting of faces disjoint from a full subcomplex A . Then there is a homotopy equivalence X ( A ) ≃ X − A. Proposition 2.2.
Let X be a simplicial complex of dimension d , which is not a d -simplex.If X removed any vertex is n -acyclic, then it is 1-complementary min { d − , n } -acyclic.Proof. Let σ be any simplex of X . Since X is not a simplex, X ( σ ) = ∅ . Moreover, sinceevery simplex is a full subcomplex, it follows from Lemma 2.1 that X ( σ ) ≃ X − σ . Clearly,the contraction of σ to its vertex extends to give a homotopy equivalence between X − σ and X removed a vertex. Then X ( σ ) is n -acyclic, completing the proof. (cid:3) Since a sphere removed a point is contractible, we have:
Corollary 2.3.
Every simplicial d -sphere is 1-complementary ( d − -acyclic. We consider the complementary acyclicity of simplicial spheres further. Recall that thecohomological dimension of a space A (over Z ), denoted by cd( A ), is defined to be thegreatest integer n such that e H n ( A ) = 0, where we put cd( A ) = − A is acyclic. Clearly,the cohomological dimension is at most the homotopy dimension. If A is a subcomplex ofa d -sphere triangulation X , then by Alexander duality, there is an isomorphism for each i e H d − i − ( X − A ) ∼ = e H i ( A ) . Hence by Lemma 2.1, X ( A ) is ( d − cd( A ) − Proposition 2.4.
Let X be a d -sphere triangulation. Given integers r ≥ and n ≥ − ,suppose that every subcomplex A of X with faces σ , . . . , σ k for k ≤ r and dim σ + · · · +dim σ k ≤ d − n − is of cohomological dimension at most n . Then X is r -complementary ( d − n − -acyclic. Let X be a simplicial complex. For a simplex σ of X , let N ( σ ) = X − X ( σ ). Then byLemma 2.1, N ( σ ) is contractible. We say that X is r -saturated if for any simplices σ , . . . , σ k of X with k ≤ r , N ( σ ) ∩ · · · ∩ N ( σ k ) is contractible or empty. SHO HASUI, DAISUKE KISHIMOTO, MASAHIRO TAKEDA, AND MITSUNOBU TSUTAYA
Corollary 2.5.
Every r -saturated simplicial d -sphere is r -complementary ( d − r ) -acyclic.Proof. Let X be an r -saturated simplicial d -sphere, and let A = N ( σ ) ∪ · · · ∪ N ( σ k ) forsimplices σ , . . . , σ k of X . By the nerve theorem, A is homotopy equivalent to the nerve of { N ( σ i ) ∩ · · · ∩ N ( σ i l ) } ≤ i < ···
Figure 1.
Equivariant set-up
We connect Theorem 1.3 to a group action in the standard manner as in [3, 6]. Let X bea regular CW complex. For a positive integers r , we define the discrete configuration spaceConf r ( X ) as a subcomplex of the direct product X r consisting of faces σ × · · · × σ r of X r , where σ , . . . , σ r are pairwise disjoint faces of X . Note that the canonical action of thesymmetric group Σ r on X r restricts to Conf r ( X ).Let the symmetric group Σ r act on ( R d ) r by permutation of R d . Then the fixed point setof this action is the diagonal set∆ = { ( x , . . . , x r ) ∈ ( R d ) r | x = · · · = x r } . Clearly, ( R d ) r − ∆ is homotopy equivalent to S d ( r − − . The following lemma (cf. [6,Theorem 3.9]) is the so-called configuration space–test map scheme . Consult [14] and thereferences therein for more on this scheme. Lemma 3.1.
Let X be a regular CW complex. If there is a map f : X → R d such that f ( σ ) ∩ · · · ∩ f ( σ r ) = ∅ for every pairwise disjoint faces σ , . . . , σ r of X , then there is a Σ r -equivariant map ¯ f : Conf r ( X ) → ( R d ) r − ∆ . Proof.
The map ¯ f is given by the restriction of the direct product f r : X r → ( R d ) r of r copies of f . (cid:3) VERBERG’S THEOREM FOR CELL COMPLEXES 5
Let r = p n for a prime p . The action of ( Z /p ) n on ( Z /p ) n itself given by( Z /p ) n × ( Z /p ) n → ( Z /p ) n , ( x, y ) x + y is faithful, so we get an embedding ( Z /p ) n → Σ r . In particular, we get actions of ( Z /p ) n on Conf r ( X ) and ( R d ) r . The following Borsuk-Ulam type result is proved by Blagojevi´cand Ziegler [6, Proof of Theorem 3.11] when Y is of the homotopy type of a d ( r − Y is ( d ( r − − Proposition 3.2. If Y is a ( d ( r − − -acyclic ( Z /p ) n -space where r = p n , then there isno ( Z /p ) n -equivariant map Y → ( R d ) r − ∆ . By Lemma 3.1 and Proposition 3.2, we get:
Corollary 3.3.
Let X be a regular CW complex such that Conf r ( X ) is ( d ( r − − -acyclic.Then there are pairwise disjoint faces σ , . . . , σ r of X such that f ( σ ) ∩ · · · ∩ f ( σ r ) = ∅ . Proof of Theorem 1.3
By Corollary 3.3, for proving Theorem 1.3 it remains to show that Conf r ( X ) is non-emptyand n -acyclic whenever X is ( r − n -acyclic. To this end, we describeConf r ( X ) as a homotopy colimit by modifying the description of Conf r (∆ n ) in [12, Theorem15]. We will construct a spectral sequence computing the homology of a homotopy colimitfrom a poset, which is essentially the same as the Bousfield-Kan spectral sequence if theunderlying poset is the face poset of a regular CW complex. Then we compute the acyclicityof Conf r ( X ) by this spectral sequence.4.1. Homotopy colimit.
First, we recall from [19] the definition of the homotopy colimitof a functor from a poset. Let P be a poset and F : P → Top be a functor, where weunderstand a poset P as a category such that x ≥ y in P is the unique morphism x → y .Let ∆( P ) denote the order complex of a poset P . For x < y ∈ P , let ι x,y : ∆( P ≤ x ) → ∆( P ≤ y )denote the inclusion. The homotopy colimit hocolim F is defined as the coequalizer of themaps f, g : a x Lemma 4.1. Let X be a regular CW complex and let P denote the face poset of X . Thenthere is a homeomorphism ∆( P ) ∼ = −→ X which restricts to ∆( P ≤ σ ) ∼ = −→ σ for each face σ of X . Let X be a regular CW complex, and let P denote the face poset of X . For σ < τ ∈ P , let θ σ,τ : Conf r − ( X ( τ )) → Conf r − ( X ( σ )) and ι σ,τ : σ → τ denote the inclusions. Then Conf r ( X ) is given by the coequalizer of the two maps f, g : a σ<τ ∈ P σ × Conf r − ( X ( τ )) → a σ ∈ P σ × Conf r − ( X ( σ ))where f = a σ<τ ∈ P σ × θ σ,τ and g = a σ<τ ∈ P ι σ,τ × Conf r − ( X ( τ )) . Define a functor F : P → Top by F ( σ ) = Conf r − ( X ( σ )) and F ( σ > τ ) = θ σ,τ . Then by Lemma 4.1 we get: Proposition 4.2. Conf r ( X ) = hocolim F . Spectral sequence. We construct a spectral sequence computing the homology of ahomotopy colimit of a functor from a poset. Let P be a poset. Then we have(4.1) ∆( P ) = [ x ∈ P ∆( P ≤ x ) . Let P n denote the union of all ∆( P ≤ x ) with dim ∆( P ≤ x ) ≤ n . Then we get a filtration(4.2) P ⊂ · · · ⊂ P n ⊂ P n +1 ⊂ · · · of P . Let F : P → Top be a functor. By definition, there is a projection π : hocolim F → ∆( P ), so we get a filtration π − ( P ) ⊂ · · · ⊂ π − ( P n ) ⊂ π − ( P n +1 ) ⊂ · · · of hocolim F . Then the homology spectral sequence associated to this filtration computesthe homology of hocolim F .We compute the E -term of the spectral sequence in the special case. Suppose that P is theface poset of a regular CW complex. By Lemma 4.1, the decomposition (4.1) gives a CWdecomposition of ∆( P ) such that dim ∆( P ≤ σ ) = n for dim σ = n and the filtration (4.2) isthe skeletal filtration. Then E p,q = H p + q ( π − ( P p ) , π − ( P p − )) ∼ = M dim σ = p H q ( F ( σ )) VERBERG’S THEOREM FOR CELL COMPLEXES 7 and the spectral sequence is essentially the same as the Bousfield-Kan spectral sequence [7,XII 4.5]. Summarizing, we obtain: Theorem 4.3. Let F : P → Top be a functor where P is the face poset of a regular CWcomplex. Then there is a spectral sequence E p,q ∼ = M dim σ = p H q ( F ( σ )) ⇒ H p + q (hocolim F ) . We prove the following Lemma by using the spectral sequence of Theorem 4.3. Lemma 4.4. Let P be the face poset of a regular CW complex X , and let F : P → Top bea functor. If F ( σ ) is ( n − dim σ ) -acyclic for each face σ ∈ P of dimension ≤ n + 1 , thenthere is an isomorphism for ∗ ≤ nH ∗ (hocolim F ) ∼ = H ∗ ( X ) . Proof. Let E r denote the spectral sequence of Theorem 4.3. Then for p + q ≤ nE p,q ∼ = ( q > C p (∆( P )) q = 0and E n +1 , ∼ = L k C n +1 (∆( P )) for some k ≥ 1, where C ∗ (∆( P )) denotes the cellular chaincomplex of ∆( P ) associated with the CW decomposition (4.1). By the construction ofthe spectral sequence, the differential d : E p +1 , → E p, is identified with the bound-ary map ∂ : C p +1 (∆( P )) → C p (∆( P )) for p < n and the sum of the boundary maps L k ∂ : L k C n +1 (∆( P )) → C n (∆( P )) for p = n . Then sinceIm (M k ∂ : M k C n +1 (∆( P )) → C n (∆( P )) ) = Im { ∂ : C n +1 (∆( P )) → C n (∆( P )) } we obtain for p + q ≤ n , E p,q ∼ = ( q > H p (∆( P )) q = 0so that the spectral sequence collapses at the E -term in degrees ≤ n . Thus by Lemma 4.1,the proof is complete. (cid:3) We are ready to calculate the acyclicity of Conf r ( X ). Proposition 4.5. If X is an ( r − -complementary n -acyclic regular CW complex, then Conf r ( X ) is n -acyclic.Proof. We prove Conf r ( X ) is n -acyclic by induction on r . For r = 1, Conf ( X ) = X ,which is n -acyclic by assumption. Assume that Conf r − ( Y ) is n -acyclic for any ( r − n -acyclic space Y . Consider the functor F in Proposition 4.2. Then F is a SHO HASUI, DAISUKE KISHIMOTO, MASAHIRO TAKEDA, AND MITSUNOBU TSUTAYA functor from the face poset of X and F ( σ ) = Conf r − ( X ( σ ))for each face σ of X . Since X ( σ ) is ( r − n − dim σ )-acyclic for dim σ ≤ n + 1, it follows from the induction assumption that F ( σ ) is ( n − dim σ )-acyclic for dim σ ≤ n + 1. Then by Lemmas 4.1, 4.4 and Proposition 4.2, Conf r ( X ) is non-empty and there isan isomorphism H ∗ (Conf r ( X )) ∼ = H ∗ ( X )for ∗ ≤ n . Thus since X is n -acyclic, Conf r ( X ) is n -acyclic, completing the proof. (cid:3) Now we are ready to prove Theorem 1.3. Proof of Theorem 1.3. Combine Corollary 3.3 and Proposition 4.5. (cid:3) Atomicity Theorem 1.3 shows that the Tverberg property is possessed not only by a simplex but alsoby a variety of CW complexes. But the Tverberg property of some CW complexes can bededuced from the Tverberg property of other complexes. For example, as mentioned inSection 1, the Tverberg property of a polytopal sphere is deduced from the simplex case.Then we need to identify a CW complex having the Tverberg property that is not deducedfrom other CW complexes.We say that a regular CW complex X is ( d, r ) -Tverberg if for any continuous map f : X → R d , there are pairwise disjoint faces σ , . . . , σ r of X such that f ( σ ) ∩ · · · ∩ f ( σ r ) = ∅ . Let X be a ( d, r )-Tverberg complex, and let Y be a regular CW complex. We have thefollowing two preservation of ( d, r )-Tverbergness:(1) if Y includes X as a subcomplex, then Y is ( d, r )-Tverberg;(2) if Y is a refinement of X , i.e. X = Y as a space and each face of X is a union offaces of Y , then Y is ( d, r )-Tverberg.Then we define that a finite ( d, r )-Tverberg complex is atomic if it is not a proper refine-ment of a ( d, r )-Tverberg complex and has no proper ( d, r )-Tverberg subcomplex. It isfundamental to count atomic ( d, r )-Tverberg complexes of a given topological type. Problem 5.1. Is the number of atomic ( d, r )-Tverberg complexes of a given topologicaltype finite or infinite?In dimension one, we can give a complete list of atomic (1 , , C n denote a cycle graph with VERBERG’S THEOREM FOR CELL COMPLEXES 9 n vertices for n ≥ 3. Then C is (1 , Y be the following Y-shaped graph. tt tt ✚✚❩❩ Then it is easy to see that Y is (1 , Proposition 5.2. Atomic (1 , -Tverberg complexes of dimension one are C and Y .Proof. Clearly, any proper subcomplex of C and Y are not (1 , X be afinite connected regular CW complex of dimension one. Then X is a finite simple graph. If X has a vertex of degree ≥ 3, then X includes Y as a subgraph. If every vertex of X is ofdegree ≤ 2, then X is a path graph or includes a cycle C n for some n ≥ 3. A path graph isnot (1 , R , and C n is a refinement of C for n ≥ 3. Thusthe statement is proved. (cid:3) Let Y be the Y-shaped graph as above. If we remove the center, then Y becomes discon-nected, so Y is not 1-complementary 0-acyclic. Hence Theorem 1.3 for d = 1 and r = 2 isnot applicable to a graph Y . Even worse, the equivariant method above does not seem towork for Y because Conf ( Y ) is the boundary of a hexagon. However, as in Section 4, wecan easily see that Y is (1 , , Proposition 5.3. The only atomic (2 , -Tverberg polyhedral sphere is the boundary of ∆ .Proof. By Steinitz’s theorem, any polyhedral 2-sphere is polytopal, and Gr¨unbaum [10, p.200] showed that any polytopal sphere is a refinement of the boundary of a simplex. Onthe other hand, the boundary of a 3-simplex is (2 , , (cid:3) The 2-sphere of Figure 2 is 1-complementary 1-acyclic, so (2 , , Problem 5.4. Is the number of atomic (2 , References [1] E.G. Bajm´oczy and I. B´ar´any, On a common generalization of Borsuk’s and Radon’s theorem, ActaMathematica Hungarica, (1979), 347-350.[2] I. B´ar´any, G. Kalai, and R. Meshulam, A Tverberg type theorem for matroids, A journey throughdiscrete mathematics, 115-121, Springer, Cham, 2017.[3] I. B´ar´any and P. Sober´on, Tverberg’s theorem is 50 years old: A survey, Bull. Amer. Math. Soc. (2018), no. 4, 459-492.[4] I. B´ar´any, S. B. Shlosman, and A. Sz˝ucs, On a topological generalization of a theorem of Tverberg, J.London Math. Soc. (2) (1981), no. 1, 158-164.[5] P.V.M. Blagojevi´c, A. Haase, and G.M. 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Institute of Mathematics, University of Tsukuba, 305-8571, Japan Email address : [email protected] Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan Email address : [email protected] Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan Email address : [email protected] Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan Email address ::