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Tverberg's Theorem for Cell Complexes

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Tverberg's Theorem for Cell Complexes 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 (d+1)(r−1) d (d+1)(r−1) f : ∆ → R , there are pairwise disjoint faces σ1,...,σr of ∆ such that f(σ1) ∩···∩ f(σr) =6 ∅ whenever r is a prime power. We generalize this theorem to a continuous map from a certain CW complex into a Euclidean space. 1. Introduction Let d ≥ 1 and r ≥ 2 be integers. Tverberg’s theorem states that for any given (d+1)(r−1)+1 points in Rd, 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 − 1)-simplex into Rd. The topological Tverberg theorem replaces an affine map in Tverberg’s theorem by a continuous maps: for any continuous map f : ∆(d+1)(r−1) → Rd, there are pairwise (d+1)(r−1) disjoint faces σ1,...,σr of the simplex ∆ such that f(σ1) ∩···∩ f(σr) 6= ∅ 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 a prime 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: arXiv:2101.10596v1 [math.AT] 26 Jan 2021 Problem 1.1. Can we replace a simplex in the topological Tverberg theorem by other CW complexes? There was a relevant problem posed by Tverberg [9]: can we replace a simplex by a polytope in the topological Tverberg theorem? This is affirmatively solved because the boundary of any 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. The problem was also studied by B´ar´any, Kalai and Meshulam[2] and Blagojevi´c, Haase and Ziegler [5] for matroid complexes. 2010 Mathematics Subject Classification. 52A37, 55R80. Key words and phrases. topological Tverberg theorem, discretized configuration space, homotopy colimit. 1 2 SHO HASUI, DAISUKE KISHIMOTO, MASAHIRO TAKEDA, AND MITSUNOBU TSUTAYA We will prove a simplex in the topological Tverberg theorem can be replaced by the following CW complex, where we will see the result is not a consequence of the topological Tverberg theorem unlikely to the case of polytopes. To define such a CW complex, we set notation and terminology. A face of a CW complex will mean a closed cell. For pairwise disjoint faces σ1,...,σk of a CW complex X, let X(σ1,...,σk) denote a subcomplex of X consisting of faces disjoint from σ1,...,σk. A space Y is called n-acyclic if Y is non-empty and H∗(Y ) = 0 for ∗ ≤ n, where n ≥ 0. Clearly, n-connected spaces are n-acyclic. A non-empty space is called a (−1)-acyclic. e Definition 1.2. We say that a regular CW complex X is r-complementary n-acyclic if X(σ1,...,σk) is non-empty and (n−dim σ1 −···−dim σk)-acyclic for each pairwise disjoint faces σ1,...,σk with dim σ1 + · · · + dim σk ≤ n + 1, where k = 0, 1, . , r. Examples of r-complementary n-acyclic complexes will be given in Section 2. Now we state the main theorem. Theorem 1.3. Let r be a prime power. If X is an (r − 1)-complementary (d(r − 1) − 1)- acyclic regular CW complex, then for any continuous map f : X → Rd, there are pairwise disjoint faces σ1,...,σr of X such that f(σ1) ∩···∩ f(σr) 6= ∅. 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, so in the topological Radon theorem too. We will see in Corollary 2.3 below that every sim- plicial d-sphere is 1-complementary (d − 1)-acyclic. Then we obtain a generalization of the topological Radon theorem, which is interesting because neither convexity nor antipodality of a simplicial sphere is demanded. Corollary 1.4. Let X be a simplicial d-sphere. Then for any continuous map f : X → Rd there are disjoint faces σ1,σ2 of X such that f(σ1) ∩ f(σ2) 6= ∅. 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 be non-polytopal. Thus Corollary 1.4 does not follow from the topological Radon theorem, so Theorem 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 KAKENHI 19K14535. TVERBERG’STHEOREMFORCELLCOMPLEXES 3 2. Examples This section gives examples of r-complementary n-acyclic complexes. We start with a simplex: a d-simplex and its boundary are (r − 1)-complementary (d − r)-acyclic. Then we see 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 full subcomplex of X over I is defined to be the subcomplex of X consisting of all faces whose vertices 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 − 1,n}-acyclic. Proof. Let σ be any simplex of X. Since X is not a simplex, X(σ) 6= ∅. Moreover, since every 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. Since a sphere removed a point is contractible, we have: Corollary 2.3. Every simplicial d-sphere is 1-complementary (d − 1)-acyclic. We consider the complementary acyclicity of simplicial spheres further. Recall that the cohomological dimension of a space A (over Z), denoted by cd(A), is defined to be the greatest integer n such that Hn(A) 6= 0, where we put cd(A)= −1 if A is acyclic. Clearly, the cohomological dimension is at most the homotopy dimension. If A is a subcomplex of a d-sphere triangulation X, thene by Alexander duality, there is an isomorphism for each i ∼ i Hd−i−1(X − A) = H (A). Hence by Lemma 2.1, X(A) is (d − cd(A) − 2)-acyclic, so we get: e e Proposition 2.4. Let X be a d-sphere triangulation. Given integers r ≥ 1 and n ≥ −1, suppose that every subcomplex A of X with faces σ1,...,σk for k ≤ r and dim σ1 + · · · + dim σk ≤ d − n − 1 is of cohomological dimension at most n. Then X is r-complementary (d − n − 2)-acyclic. Let X be a simplicial complex. For a simplex σ of X, let N(σ) = X − X(σ). Then by Lemma 2.1, N(σ) is contractible. We say that X is r-saturated if for any simplices σ1,...,σk of X with k ≤ r, N(σ1) ∩···∩ N(σk) is contractible or empty. 4 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(σ1) ∪···∪ N(σk) for simplices σ1,...,σk of X. By the nerve theorem, A is homotopy equivalent to the nerve of {N(σi1 ) ∩···∩ N(σil )}1≤i1<···<il≤k, implying A is of homotopy dimension at most k − 2. Thus the proof is complete by Proposition 2.4. Finally, we give a non-polyhedral example which will be considered in Section 5. Figure 1. 1-complementary 1-acyclic non-polyhedral 2-sphere 3. Equivariant set-up We connect Theorem 1.3 to a group action in the standard manner as in [3, 6]. Let X be a regular CW complex. For a positive integers r, we define the discrete configuration space r Confr(X) as a subcomplex of the direct product X consisting of faces σ1 ×···× σr of r X , where σ1,...,σr are pairwise disjoint faces of X. Note that the canonical action of the r symmetric group Σr on X restricts to Confr(X). d r d Let the symmetric group Σr act on (R ) by permutation of R . Then the fixed point set of this action is the diagonal set d r ∆= {(x1,...,xr) ∈ (R ) | x1 = · · · = xr}. Clearly, (Rd)r − ∆ is homotopy equivalent to Sd(r−1)−1. The following lemma (cf. [6, Theorem 3.9]) is the so-called configuration space–test map scheme. Consult [14] and the references therein for more on this scheme. Lemma 3.1. Let X be a regular CW complex. If there is a map f : X → Rd such that f(σ1) ∩···∩ f(σr) = ∅ for every pairwise disjoint faces σ1,...,σr of X, then there is a Σr-equivariant map d r f¯: Confr(X) → (R ) − ∆.
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