Tverberg's Theorem for Cell Complexes

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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 simplicial 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 boundary 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 supported 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<···

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 ) − ∆.

Proof. The map f¯ is given by the restriction of the direct product f r : Xr → (Rd)r of r copies of f. TVERBERG’STHEOREMFORCELLCOMPLEXES 5

Let r = pn 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) 7→ x + y n n is faithful, so we get an embedding (Z/p) → Σr. In particular, we get actions of (Z/p) d r on Confr(X) and (R ) . The following Borsuk-Ulam type result is proved by Blagojevi´c and Ziegler [6, Proof of Theorem 3.11] when Y is of the homotopy type of a d(r − 1)- sphere. We can easily see that the proof of Blagojevi´cand Ziegler works verbatim when Y is (d(r − 1) − 1)-acyclic.

Proposition 3.2. If Y is a (d(r − 1) − 1)-acyclic (Z/p)n-space where r = pn, then there is no (Z/p)n-equivariant map Y → (Rd)r − ∆.

By Lemma 3.1 and Proposition 3.2, we get:

Corollary 3.3. Let X be a regular CW complex such that Confr(X) is (d(r−1)−1)-acyclic. Then there are pairwise disjoint faces σ1,...,σr of X such that

f(σ1) ∩···∩ f(σr) 6= ∅.

4. Proof of Theorem 1.3

By Corollary 3.3, for proving Theorem 1.3 it remains to show that Confr(X) is non-empty and n-acyclic whenever X is (r − 1)-complementary n-acyclic. To this end, we describe n Confr(X) as a homotopy colimit by modifying the description of Confr(∆ ) in [12, Theorem 15]. We will construct a spectral sequence computing the homology of a homotopy colimit from a poset, which is essentially the same as the Bousﬁeld-Kan spectral sequence if the underlying poset is the face poset of a regular CW complex. Then we compute the acyclicity of Confr(X) by this spectral sequence.

4.1. Homotopy colimit. First, we recall from [19] the deﬁnition of the homotopy colimit of a functor from a poset. Let P be a poset and F : P → Top be a functor, where we understand 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

f, g : ∆(P≤x) × F (y) → ∆(P≤x) × F (x) x

f = 1∆(P≤x) × F (y>x) and g = ιx,y × 1F (y). x

Lemma 4.1. Let X be a regular CW complex and let P denote the face poset of X. Then there 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

θσ,τ : Confr−1(X(τ)) → Confr−1(X(σ)) and ισ,τ : σ → τ denote the inclusions. Then Confr(X) is given by the coequalizer of the two maps

f, g : σ × Confr−1(X(τ)) → σ × Confr−1(X(σ)) σ<τ∈P σ∈P a a where

f = 1σ × θσ,τ and g = ισ,τ × 1Confr−1(X(τ)). σ<τ∈P σ<τ∈P a a Deﬁne a functor F : P → Top by

F (σ) = Confr−1(X(σ)) and F (σ>τ)= θσ,τ . Then by Lemma 4.1 we get:

Proposition 4.2. Confr(X) = hocolim F .

4.2. Spectral sequence. We construct a spectral sequence computing the homology of a homotopy colimit of a functor from a poset. Let P be a poset. Then we have

(4.1) ∆(P )= ∆(P≤x). x∈P [ Let Pn denote the union of all ∆(P≤x) with dim ∆(P≤x) ≤ n. Then we get a ﬁltration

(4.2) P0 ⊂···⊂ Pn ⊂ Pn+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 −1 −1 −1 π (P0) ⊂···⊂ π (Pn) ⊂ π (Pn+1) ⊂··· of hocolim F . Then the homology spectral sequence associated to this filtration computes the homology of hocolim F . We compute the E1-term of the spectral sequence in the special case. Suppose that P is the face poset of a regular CW complex. By Lemma 4.1, the decomposition (4.1) gives a CW decomposition of ∆(P ) such that dim ∆(P≤σ)= n for dim σ = n and the filtration (4.2) is the skeletal filtration. Then 1 −1 −1 ∼ Ep,q = Hp+q(π (Pp),π (Pp−1)) = Hq(F (σ)) dimMσ=p TVERBERG’STHEOREMFORCELLCOMPLEXES 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 CW complex. Then there is a spectral sequence 1 ∼ Ep,q = Hq(F (σ)) ⇒ Hp+q(hocolim F ). dimMσ=p 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 be a functor. If F (σ) is (n − dim σ)-acyclic for each face σ ∈ P of dimension ≤ n + 1, then there is an isomorphism for ∗≤ n

H∗(hocolim F ) =∼ H∗(X).

Proof. Let Er denote the spectral sequence of Theorem 4.3. Then for p + q ≤ n

1 ∼ 0 q > 0 Ep,q = (Cp(∆(P )) q = 0 1 ∼ and En+1,0 = k Cn+1(∆(P )) for some k ≥ 1, where C∗(∆(P )) denotes the cellular chain complex of ∆(P ) associated with the CW decomposition (4.1). By the construction of L 1 1 1 the spectral sequence, the diﬀerential d : Ep+1,0 → Ep,0 is identiﬁed with the boundary map ∂ : Cp+1(∆(P )) → Cp(∆(P )) for p< n and the sum of the boundary maps

k ∂ : k Cn+1(∆(P )) → Cn(∆(P )) for p = n. Then since L L Im ∂ : Cn+1(∆(P )) → Cn(∆(P )) = Im{∂ : Cn+1(∆(P )) → Cn(∆(P ))} ( k k ) M M we obtain for p + q ≤ n,

2 ∼ 0 q > 0 Ep,q = (Hp(∆(P )) q = 0 so that the spectral sequence collapses at the E2-term in degrees ≤ n. Thus by Lemma 4.1, the proof is complete.

We are ready to calculate the acyclicity of Confr(X).

Proposition 4.5. If X is an (r − 1)-complementary n-acyclic regular CW complex, then Confr(X) is n-acyclic.

Proof. We prove Confr(X) is n-acyclic by induction on r. For r = 1, Conf1(X) = X, which is n-acyclic by assumption. Assume that Confr−1(Y ) is n-acyclic for any (r − 2)- complementary n-acyclic space Y . Consider the functor F in Proposition 4.2. Then F is a 8 SHO HASUI, DAISUKE KISHIMOTO, MASAHIRO TAKEDA, AND MITSUNOBU TSUTAYA functor from the face poset of X and

F (σ) = Confr−1(X(σ)) for each face σ of X. Since X(σ) is (r − 2)-complementary (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, Confr(X) is non-empty and there is an isomorphism

H∗(Confr(X)) =∼ H∗(X) for ∗≤ n. Thus since X is n-acyclic, Confr(X) is n-acyclic, completing the proof.

Now we are ready to prove Theorem 1.3.

Proof of Theorem 1.3. Combine Corollary 3.3 and Proposition 4.5.

5. Atomicity

Theorem 1.3 shows that the Tverberg property is possessed not only by a simplex but also by a variety of CW complexes. But the Tverberg property of some CW complexes can be deduced from the Tverberg property of other complexes. For example, as mentioned in Section 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 deduced from other CW complexes. We say that a regular CW complex X is (d, r)-Tverberg if for any continuous map f : X → d R , there are pairwise disjoint faces σ1,...,σr of X such that

f(σ1) ∩···∩ f(σr) 6= ∅. Let X be a (d, r)-Tverberg complex, and let Y be a regular CW complex. We have the following 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 of faces 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 refinement of a (d, r)-Tverberg complex and has no proper (d, r)-Tverberg subcomplex. It is fundamental 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 topological type ﬁnite or inﬁnite?

In dimension one, we can give a complete list of atomic (1, 2)-Tverberg complexes, where only (1, 2)-Tverbergness is non-trivial in dimension one. Let Cn denote a cycle graph with TVERBERG’STHEOREMFORCELLCOMPLEXES 9 n vertices for n ≥ 3. Then C3 is (1, 2)-Tverberg by the topological Tverberg theorem. Let Y be the following Y-shaped graph. t t ❩ ✚ ❩t✚

Then it is easy to see that Y is (1, 2)-Tverberg.

Proposition 5.2. Atomic (1, 2)-Tverberg complexes of dimension one are C3 and Y .

Proof. Clearly, any proper subcomplex of C3 and Y are not (1, 2)-Tverberg. Let X be a finite 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 of degree ≤ 2, then X is a path graph or includes a cycle Cn for some n ≥ 3. A path graph is not (1, 2)-Tverberg because it embeds into R, and Cn is a refinement of C3 for n ≥ 3. Thus the statement is proved.

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 is not applicable to a graph Y . Even worse, the equivariant method above does not seem to work for Y because Conf2(Y ) is the boundary of a hexagon. However, as in Section 4, we can easily see that Y is (1, 2)-Tverberg. Then there should be a method for generalizing the topological Tverberg theorem to CW complexes including the Y-shaped graph. One possibility is the Fadell-Husseini index which is used by Blagojevi´c, Haase and Ziegler [5] in the study of the topological Tverberg theorem for matroids. In dimension two, we can also identify a (2, 2)-Tverberg polyhedral sphere.

Proposition 5.3. The only atomic (2, 2)-Tverberg polyhedral sphere is the boundary of ∆3.

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. On the other hand, the boundary of a 3-simplex is (2, 2)-Tverberg by the topological Tverberg theorem, and any of its proper subcomplex is not (2, 2)-Tverberg, completing the proof.

The 2-sphere of Figure 2 is 1-complementary 1-acyclic, so (2, 2)-Tverberg by Theorem 1.3. Moreover, it is atomic and non-polyhedral. Then there is a non-polyhedral atomic (2, 2)- Tverberg 2-sphere. Now we pose a weaker version of Problem 5.1.

Problem 5.4. Is the number of atomic (2, 2)-Tverberg 2-spheres ﬁnite or inﬁnite? 10 SHO HASUI, DAISUKE KISHIMOTO, MASAHIRO TAKEDA, AND MITSUNOBU TSUTAYA

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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]

Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan Email address: [email protected]