HASUI, SHO

KISHIMOTO, DAISUKE

TAKEDA, MASAHIRO

TSUTAYA, MITSUNOBU

English

Kyoto University Research Information Repository

RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:TVERBERG'S THEOREM FOR CELLCOMPLEXES (New trends oftransformation groups)HASUI, SHO; KISHIMOTO, DAISUKE; TAKEDA,MASAHIRO; TSUTAYA, MITSUNOBUHASUI, SHO ...[et al]. TVERBERG'S THEOREM FOR CELL COMPLEXES (New trends oftransformation groups). 数理解析研究所講究録 2022, 2231: 129-1342022-11http://hdl.handle.net/2433/279790129TVERBERG'S THEOREM FOR CELL COMPLEXES SHO RASUi, DAISUKE KISHIMOTO, MASAHIRO TAKEDA, AND M. TSUTAYA 1. INTRODUCTION This is a survey of the paper [13]. Radon's theorem on configurations of points in the Euclidean space states that given and d + 2 points in配， wecan partition these points into two subsets whose convex hulls have a point in common. It is natural to consider more points and more subsets, and Tverberg [17] generalized Radon's theorem along this direction: any given (d+ l)(r-1) + 1 points in配 canbe a partitioned into r disjoint subsets whose convex hulls have a point in common. It is useful to translate Tverberg's theorem in terms of an affine map: given any affine map f:△（d+l)(r-1)→配， thereare pairwise disjoint faces u1,...ぶ of△(d+l)(r-l)such that f（び1),・ ・ ・, f(びr)have a point in common. This theorem has been of great interest in combinatorics for over 50 years, and a variety of its generalization have been obtained. See comprehensive surveys [1, 3, 6] for history and developments around Tverberg's theorem. Now we consider a topological generalization. Question 1.1. What happens if a map△(d+l)(r-1)→配 isnot affine but only continuous? Here is an answer to this question, which is now called the topological Tverberg theorem. Theorem 1.2. If r is a prime power, then for any continuous map f:△（d+l)(r-1)→ 訊 thereare pairwise disjoint facesび1,・・・,urof△(d+l)(r-l) such that f（叫...'f（叫 havea point in common. Remarks on the topological Tverberg theorem are in order. The topological Tverber苔theoremwas proved by Barany, Shlosman and Szucs [4] when r is a prime, and by Ozaydin [16] and Volovoikov [18] when r is a prime power. As long as we look at the proof the condition for r being a prime power seems quite technical. But Frick [9] proved that the condition that r is a prime power is necessary. Let us consider a generalization of the topological Tverberg theorem. In [10], Tverberg asked whether or not it is possible to generalize the topological Tverberg theorem to continuous maps from (d + l)(r -1)-polytopes into配． Theanswer is positive because the boundary of a convex n、-polytopeis a refinement of the boundary of an n-simplex as in [11, p. 200] and the result follows from the topolog-ical Tverberg theorem. Then Tverberg's question does not contribute to a proper generalization of the topological Tverberg theorem, and so we further ask: Question 1.3. For which CW  complexes can we generalize the topological Tver-berg theorem to continuous maps from them into Euclidean spaces? Recently, B紅紐y,Kalai and Meshulam [2] and Blagojevic, Haase and Ziegler [5] constructed affirmative examples of matroid complexes for Question 1.3 in a purely Date: June 6, 2022. 130S. HASUI, D. KISHIMOTO, M. TAKEDA, AND M. TSUTAYA combinatorial way. In this survey, we give a new affirmative class of regular CW  complexes from a topological point of view. To state the main theorem, we set notation and terminology. Let X be a regular CW  complex. A face of X means its closed cel. For faces 0-1,..., O"k of X, let X（吐・..四） denotethe subcomplex of X consisting of faces which do not intersect with o-1,...，びk・ Recall that a space Y is called n-acyclic if几(Y)= 0 for*さn.For convenience, a non-empty space will be called (-1)-acyclic, so that any n-acyclic space for n :> 0 will be assumed non-empty. We define a regular CW  complex that we are going to consider in this paper. Definition 1.4. We say that a regular CW  complex X is k-complementary n-acyclic if X (0-1,...，叫 is(n-dimび1-• • ・ -dim o-i)-acyclic for any pairwise disjoint facesび1,...,O"i of X such that dimび1+ ・ ・ ・ + dimo-i::; n +land 0さiさk.Now we state the main theorem. Theorem 1.5. If X is an (r -l)-complementary (d(r-l)-l)-acyclic regular CW  complex and r is a prime power, then for any continuous map f: X →配， thereare pairwise disjoint facesび1'...，年。fX such that f(o-1),..., f（叫 havea point zn common. Since a (d + 1)-simplex is k-complementary (d -k)-acyclic for 1 <: k <:'. d + l, the topological Tverberg theorem is recovered by Theorem 1.5. Moreover, we can prove: Proposition 1.6. Every simplicial d-sphere is k-complementary (d-k)-acyclicfor l,Sk,Sd+l. Then we get: Corollary 1. 7. If S is a simplicial ((d + l) (r -l) -1)-sphere and r is a prime power, then for any continuous map f: S→配， thereare pai加 isedisjoint faces u1,...，年。fS such that f(u1),..., f(ur) have a point in common. Griinbaum and Sreedharan [12] constructed a simplicial 3-sphere which is not polytopal. Moreover, Kalai [14] proved that ford large, "most" simplicial d-spheres are not polytopal. Then Corollary 1.7, hence Theorem 1.5 is a substantial gen-eralization of the topological Tverberg theorem. We began with a property of a configuration of points in the Euclidean space and ended up with a property of a simplicial sphere. 2. SKETCH OF THE PROOF Let X be a regular CW  complex. The discretized configuration space Confr(X) is defined as the subcomplex of the direct product xr consisting of faces <71 x • • • x巧such thatび1,...,びrare pairwise disjoint faces of X. The discretized configuration space is often called the deleted product in combinatorics, alternatively. Let△ = {(xi,...,叫） E （配）rI X1 =... =叫｝． Thereis a homotopy equivalence (2.1) （酎）r —△ c:c Sd(r-1)-1 _ Note that the symmetric group江 actson Confr(X) and（配）r＿△ by permuting of entries. The following lemma is proved in [6, Theorem 3.9]. Lemma 2.1. Let X be a regular C W  complex. If there is a continuous map X→配such that f（び1)n..•n/(びr) = 0 for al pai加 isedisjoint facesび1,...，びrof X, then there is a ~r-map Confr(X)→ （酎）r＿△． 131TVERBERG'S THEOREM FOR CELL COMPLEXES If r is a prime, then the actions of Z/r c江 onConfr(X) and （配）r —△ arefree. So we can apply the Borsuk-Ulam theorem to Lemma 2.1. If r is a prime power, then we can also apply a generalization of the Borsuk-Ulam theorem in [18] (cf. [7]). More precisely, we get: p roposition 2.2. Let X be a regular CW  complex such that Confr(X) is (d(r -1) -1)-acyclic. If r is a prime power, then for any continuous map f: X →記there are pairwise disjoint facesび1,...,(J"rofX suchthatf（び1),・ ・ ・,f (J"r) have a point in common. In order to compute the homology of Confr(X), we shall give its homotopy decomposition. Let P be a poset. Hereafter, we understand P as a category such that objects are elements of P and there is a unique morphism x→y for x > y E P. For x E P, let P'.:'.x = {y E PI y'.S x}. The order complex△(P) is the geometric realization of an abstract simplicial complex whose simplices are finite chains xa < x1 < ・ ・ ・＜咋 inP. Let F: P→Top be a functor. We define two maps f,g: I△(P←)x F(y)→ I △(P'.':'.x) x F(x) x<yEP のEPby f = H 1• (P~x) X F(y > x) and g = I lx,y X lF(y), x<yEP x<yEP where伍，y:△（p'.oX)→△(P'.oy) denotes the inclusion for x < y. As in [19], the ho-motopy colimit hocolim F is defined to be the coequalizer of f and g. By definition, there is a natural projection (2.2) 1r : hocolim F→ △(P). We recall a property of regular C W  complexes that we are going to use. For a CW  complex X, let P(X) denote its face poset. The following lemma is proved in [15, Theorem 1.6, Chapter II]. Lemma 2.3. Let X be a regular C W  complex. Then there is a homeomoryhism △(P(X)）ごx竺which restricts to a homeomorphism△(P(X)叫→(Jfor each face(J． Now we describe Confr(X) in terms of a homotopy colimit. Similarly to the Fad ell-Neuwirth fi bration [ 8], we consider the first projection 7r : Conf r (X)→X. Then for each face(Jof X, we have T―l (Int((J） ＝Conf口 (X((J）．Thus since X (J）C X(r) for(J ＞r, Confr(X) is obtained by gluing(JxConf口 (X((J）along the inclusions (Jx Confr-1(X(CJ)）← T x Confr-1(X((J））→ T x Confr-1(X(r)) for(J ＞ r. In other words, Confr(X) is homeomorphic to the coequalizer of two maps f,g: Il TX Confr-1(X((J））→ I T x Conf→(X(r)) Tく咋P(X) rEP(X) defined by f = I lr X知 and g = I 好，aX lconfr_i(X（a)), TくのEP(X) TくびEP(X)132S. HASUI, D. KISHIMOTO, M. TAKEDA, AND M. TSUTAYA where 0a,T: Confr-1(X(CT))→Confr-1(X(T)) and lT,a: T→CT are inclusions for び＞ T.Now we define a functor Fr: P(X)→Top by Fr（び） ＝Confr-1(X（<J) and F（び＞ T)= 0a,T・By Lemma 2.3, there is a natural homeomorphism△(P(X):c:;a)竺 CTfor each face CT of X. Then by the above observation, we get: Theorem 2.4. There is a homeomorphism Confr(X)竺 hocolimFr.Then we can apply the Bousfield-Kan spectral sequence to compute the homology ofConfr(X). However, the E1-term of the Bousfield-Kan spectral sequence includes a plenty of degenerate elements, and so we will apply the following variant of the Bousfield-Kan spectral sequence. Proposition 2.5. Let X be a regular C W  complex, and let F: P(X)→Top be a functor. Then there is a spectral sequence 尻，q 竺〶 Hq(F （(T)) ⇒ Hp+q (hocolim F). Now we get: 咋 P(X)dima=p Lemma 2.6. Let X be a regular C W  complex, and let F: P(X)→Top be a functor such that F(a) is (n -dim a) acyclic for each a E P(X) with dim a：：：いl.Then there is an isomorphism for *： n H,(hocolimF) ~ H,(X) Thus we obtain the following corollary which completes the proof of Theorem 1.5 by Proposition 2.2. Corollary 2.7. If X is an (r -l)-complementary n-acyclic regular CW  complex, then Confr(X) is n-acyclic. 3. ATOMICITY Theorem 1.5 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 that of other complexes. For example, as mentioned in Section 1, the Tverberg property of a polytopal sphere is deduced from a simplex. This section studies C W  complexes having the Tverberg property that is not induced from other C W  complexes. We say that a regular C W  complex X is (d, r)-Tverberg if for any continu-ous map f: X →配， thereare pairwise disjoint facesび1,..．心 ofX such that f（叫，．．．，f(叫 havea point in common. For example, by Theorem 1.5, (r -1)-complementary (d(r-1)-1)-acyclic regular CW  complexes are (d, r)-Tverberg. Let X be a (d, r)-Tverberg regular CW  complex. Observe that a regular CW  complex Y is (d, r)-Tverberg if either of the following conditions is satisfied: (1) Xis a subcomplex of Y; (2) Y is a refinement of X, that is, X 竺 Yand each face of X is the union of faces of Y. This observation leads us to: Definition 3.1. A (d, r)-Tverberg regular CW  complex is called atomic if it does not include a proper subcomplex which is (d, r)-Tverberg or it is not a refinement of a proper (d, r)-Tverberg complex. 133TVERBERG'S THEOREM FOR CELL COMPLEXES Here is a fundamental problem on (d, r)-Tverberg complexes. Problem 3.2. Given d, r and n, are there only finitely many atomic (d, r)-Tverberg finite complexes of dimension n? First, we consider 1-dimensional (1, 2)-Tverberg finite complexes. Let Cn denote the cycle graph with n vertices for n ~ 3. Then by Corollary 1.7, Cn is (1, 2)-Tverberg. Let Y be the Y-shaped graph depicted below. Then by the intermediate value theorem, we can see that Y is (1, 2)-Tverberg too. 了Proposition 3.3. The only atomic 1-dimensional (1, 2)-Tverberg finite complexes are C3 and Y. Remark 3.4. If we remove its center vertex of Y, then it becomes disconnected. Hence Y is not 1-complementary 0-acyclic, so that we cannot apply Theorem 1.5 ford= land r = 2 to deduce that Y is (1, 2)-Tverberg. However, we can directly see wgt2;2 (Conむ(Y))= 1, implying that Y is (1, 2)-Tverberg, because Conむ(Y)is a hexagon so that Lemma ?? applies. Next, we consider (2, 2)-Tverberg polyhedral 2-spheres. Let 8幻 denotethe boundary of an n-simplex. Proposition 3.5. The only atomic (2, 2)-Tverberg polyhedral 2-sphere is 8△釘The following 2-sphere is an atomic (2, 2)-Tverberg complex, and so there may be other atomic (2, 2)-Tverberg 2-spheres which are not polyhedral. Then we pose a problem much weaker than Problem 3.2 but stil interesting. Problem 3.6. Are there only finitely many atomic (2, 2)-Tverberg 2-spheres? REFERENCES [1] I. B紅any,P.V.M. Blagojevic, and G.M. Ziegler, Tverberg's theorem at 50: extensions and counterexamples, Notices Arner. Math. Soc. 73 (2016), no. 3, 732-739. 1 [2] I. Barany, G. Kalai, and R. Meshularn, A Tverberg type theorem for rnatroids, A journey through discrete mathematics, pp. 115-121, Springer, Cham, 2017. 1 [3] I. Barany and P. Soberon, Tverberg's theorem is 50 years old: A survey, Bull. Amer. Math. Soc. 55 (2018), no. 4, 459-492. 1 [4] I. B年 ny,S. B. Shlosman, and A. Sz(ics, On a topological generalization of a theorem of Tverberg, J. London Math. Soc. (2) 23 (1981), no. 1, 158-164. 1 [5] P.V.M. Blagojevic, A. Haase, and G.M. Ziegler, Tverberg-type theorems for matroids: a counterexample and a proof, Combinatorica 39 (2019), no. 3, 477-500. 1 [6] P.V.M. Blagojevic and G.M. Ziegler, Beyond the Borsuk-Ulam theorem: The topological Tverberg story, A journey through discrete mathematics, pp. 273-341, Springer, Cham, 2017. 1, 2 134S. HASUI, D. KISHIMOTO, M. TAKEDA, AND M. TSUTAYA [7] E. Fadell and S. Husseini, An ideal-valued cohomological index theory with applications to Borsuk-Ulam and Bourgin-Yang theorems, Ergodic Theory Dynam. Systems, 8 (1988), 73-85. 3 [8] E. Fadell and L. Neuwirth, Configuration spaces, Math. Scand. 10 (1962), no. 4, 111-118. 3 [9] F. Frick, Counterexamples to the topological Tverberg conjecture, Oberwolfach Reports 12 (2015), no. 1, 318-321. 1 [10] P.M. Gruber and R. Schneider, Problems in geometric convexity, Contributions to geometry (Proc. Geom. Sympos., Siegen, 1978), pp. 255-278, Birkhiiuser, Basel-Boston, Mass., 1979. 1 [11] B. Griinbaum, Convex Polytopes, Graduate Texts in Mathematics 221, Springer-Verlag, New York, second edition, 2003. 1 [12] B. Griinbaum and V.P. Sreedharan, An enumeration of simplicial 4-polytopes with 8 vertices, J. Comb. Theory 2 (1967), 437-465. 2 [13] S. Hasui, D. Kishimoto, and M. Takeda, M. Tsutaya, Tverberg's theorem for cell complexes, https: //arxiv. org/abs/2101.10596. 1 [14] G. Kalai, Many triangulated spheres, Discrete Comput. Geom. 3 (1988), 1-14. 2 [15] A.T. Lundell and S. Weingram, The Topology of CW  Complexes, van Nostrand, New York, 1969. 3 [16] M. Ozaydin, Equivariant maps for the symmetric group, Unpublished preprint, University of Winsconsin-Madison, 1987. 1 [17] H. Tverberg, A generalization of Radon's theorem. I, Bull. Austral. Math. Soc. 24 (1981), no. 3, 321-325. 1 [18] A.Y. Volovikov, On a topological generalization of Tverberg's theorem, Mat. Zametki 59 (1996), no. 3, 454-456, (English transl.) Math. Notes 59 (1996), no. 3-4, 324-325. 1, 3 [19] G. Ziegler and R. Zivaljevic, Homotopy types of subspace arrangements via diagrams of spaces, Math. Ann. 295 (1993) 527-548. 3 DEPARTMENT OF MATHEMATICAL SCIENCES, OSAKA PREFECTURE UNIVERSITY, SAKAI, 599-8531, JAPAN Email address: s. hasuicms. osakafu-u. ac. jp FACULTY OF MATHEMATICS, KYUSHU UNIVERSITY, FUKUOKA, 819-0395, JAPAN Email address: kishimotocmath. kyushu-u. ac. j p DEPARTMENT OF MATHEMATICS, KYOTO UNIVERSITY, KYOTO, 606-8502, JAPAN Email address: takeda. masahiro. 87ucst. kyoto-u. ac. j p FACULTY OF MATHEMATICS, KYUSHU UNIVERSITY, FUKUOKA, 819-0395, JAPAN Email address: tsutayacmath.kyushu-u.ac.jp 

TVERBERG'S THEOREM FOR CELL COMPLEXES (New trends of transformation groups)

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TVERBERG'S THEOREM FOR CELL COMPLEXES (New trends of transformation groups)

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