Symmetric multiple chessboard complexes and some theorems of Tverberg type

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1 Symmetric multiple chessboard complexes and some theorems of Tverberg type Siniša Vrećica, University of Belgrade joint work with Duško Jojić and Rade Živaljević

2 Chessboard complex m,n vertices of m,n : squares in a chessboard which has n rows and m columns,

3 Chessboard complex m,n vertices of m,n : squares in a chessboard which has n rows and m columns, simplices of m,n : non-taking rooks placements, i.e. at most one vertex from each row and each column,

4 Chessboard complex m,n vertices of m,n : squares in a chessboard which has n rows and m columns, simplices of m,n : non-taking rooks placements, i.e. at most one vertex from each row and each column, The first examples: 3,2 is a hexagon, 4,3 is a torus.

5 The first examples Figure: 3,2 is a hexagon

6 The first examples Figure: 4,3 is a torus

7 Chessboard complexes appear as... coset complex of certain subgroups in the symmetric group

8 Chessboard complexes appear as... coset complex of certain subgroups in the symmetric group m,n = (S n, H m )

9 Chessboard complexes appear as... coset complex of certain subgroups in the symmetric group m,n = (S n, H m ) matching complex in a complete bipartite graph

10 Chessboard complexes appear as... coset complex of certain subgroups in the symmetric group m,n = (S n, H m ) matching complex in a complete bipartite graph m,n = M(K m,n )

11 Chessboard complexes appear as... coset complex of certain subgroups in the symmetric group m,n = (S n, H m ) matching complex in a complete bipartite graph m,n = M(K m,n ) n-fold 2-deleted join of vertices of the (m 1)-simplex

12 Chessboard complexes appear as... coset complex of certain subgroups in the symmetric group m,n = (S n, H m ) matching complex in a complete bipartite graph m,n = M(K m,n ) n-fold 2-deleted join of vertices of the (m 1)-simplex m,n = ( (σ m 1 ) (0)) n (2) = ( [1] m (2) ) n (2)

13 Properties of chessboard complexes are important! m,n is (n 2)-connected for m 2n 1.

14 Properties of chessboard complexes are important! m,n is (n 2)-connected for m 2n 1. For a prime-power r = p α there is no (Z p ) α -map ( r,2r 1 ) (d+1) ( R d) r.

15 Properties of chessboard complexes are important! m,n is (n 2)-connected for m 2n 1. For a prime-power r = p α there is no (Z p ) α -map ( r,2r 1 ) (d+1) ( R d) r. This property was used in the proofs of some results in Discrete Geometry and Combinatorics.

16 Multiple chessboard complex k 1,...,k n ;l 1,...,l m m,n vertices: squares in a chessboard which has n rows and m columns,

17 Multiple chessboard complex k 1,...,k n ;l 1,...,l m m,n vertices: squares in a chessboard which has n rows and m columns, simplices: having at most k i vertices from the i-th row and at most l j vertices from the j-th column for each i, j,

18 Multiple chessboard complex k 1,...,k n ;l 1,...,l m m,n vertices: squares in a chessboard which has n rows and m columns, simplices: having at most k i vertices from the i-th row and at most l j vertices from the j-th column for each i, j, Figure: k 1,...,k m;l 1,...,l m m,n

19 The important special case If k 1 = = k n = k and l 1 = = l m = l, we denote it by k;l m,n.

20 The important special case If k 1 = = k n = k and l 1 = = l m = l, we denote it by k;l m,n. We are mainly interested in k 1,...,k n;1 m,n and k;1 m,n, i.e. in the case l 1 = = l m = 1.

21 The important special case If k 1 = = k n = k and l 1 = = l m = l, we denote it by k;l m,n. We are mainly interested in k 1,...,k n;1 m,n and k;1 m,n, i.e. in the case l 1 = = l m = 1. The first examples: 2;1 3,2 S 1 D 1, 2,1;1 4,2 S 2, 2;1 5,2 S 3.

22 The new examples Figure: 2,1 3,2 is a triangulation of cylinder

23 The new examples Figure: 2,1;1 4,2 = S 2

24 2,1 5,2 S3 The new examples

25 The new examples 2,1 5,2 S3 link of a vertex in 2,1 5,2 is 2,1;1 4,2 = S 2

26 The new examples 2,1 5,2 S3 link of a vertex in 2,1 5,2 is 2,1;1 4,2 = S 2 link of an edge in 2,1 5,2 is a circle (old chessboard complex 3,2 or 2,1 3,1 )

27 The new examples 2,1 5,2 S3 link of a vertex in 2,1 5,2 is 2,1;1 4,2 = S 2 link of an edge in 2,1 5,2 is a circle (old chessboard complex 3,2 or 2,1 3,1 ) link of a 2-simplex in 2,1 5,2 is a set of two points

28 The new examples 2,1 5,2 S3 link of a vertex in 2,1 5,2 is 2,1;1 4,2 = S 2 link of an edge in 2,1 5,2 is a circle (old chessboard complex 3,2 or 2,1 3,1 ) link of a 2-simplex in 2,1 5,2 is a set of two points 2,1 5,2 is a 2-connected, simplicial 3-manifold.

29 The appearances k;l m,n is a matching complex of K m,n where each red vertex is matched with at most k blue vertices, and each blue vertex is matched with at most l red vertices.

30 The appearances k;l m,n is a matching complex of K m,n where each red vertex is matched with at most k blue vertices, and each blue vertex is matched with at most l red vertices. k;l m,n is n-fold (l + 1)-deleted join of the (k 1)-skeleton of the (m 1)-simplex or n-fold (l + 1)-deleted join of m-fold (k + 1)-deleted join of a point.

31 The appearances k;l m,n is a matching complex of K m,n where each red vertex is matched with at most k blue vertices, and each blue vertex is matched with at most l red vertices. k;l m,n is n-fold (l + 1)-deleted join of the (k 1)-skeleton of the (m 1)-simplex or n-fold (l + 1)-deleted join of m-fold (k + 1)-deleted join of a point. k;l m,n = ( (σ m 1 ) (k 1)) n (l+1) = ( [1] m (k+1) ) n (l+1)

32 The appearances k;l m,n is a matching complex of K m,n where each red vertex is matched with at most k blue vertices, and each blue vertex is matched with at most l red vertices. k;l m,n is n-fold (l + 1)-deleted join of the (k 1)-skeleton of the (m 1)-simplex or n-fold (l + 1)-deleted join of m-fold (k + 1)-deleted join of a point. k;l m,n = ( (σ m 1 ) (k 1)) n = ( ) [1] m n (l+1) (k+1) (l+1) Establishing the topological properties of these complexes was our main motivation.

33 Topological properties Theorem: Chessboard complex k 1,...,k n;1 m,n is µ-connected where µ = min{m n 1; k k n 2}.

34 Topological properties Theorem: Chessboard complex k 1,...,k n;1 m,n is µ-connected where µ = min{m n 1; k k n 2}. Corrolary: If m k k n + n 1, the complex k 1,...,k n;1 m,n is (k k n 2)-connected. In particular, if m (k + 1)n 1, then k;1 m,n is (kn 2)-connected.

35 Topological properties Theorem: Chessboard complex k 1,...,k n;1 m,n is µ-connected where µ = min{m n 1; k k n 2}. Corrolary: If m k k n + n 1, the complex k 1,...,k n;1 m,n is (k k n 2)-connected. In particular, if m (k + 1)n 1, then k;1 m,n is (kn 2)-connected. Theorem: If m k k n + n 1, the complex k 1,...,k n;1 m,n is shellable.

36 The applications A colored Tverberg type theorem with smaller number of colors, allowing k vertices of the same color in each face.

37 The applications A colored Tverberg type theorem with smaller number of colors, allowing k vertices of the same color in each face. We could consider topological Tverberg theorem and require the dimensions of faces of a simplex whose images intersect to be prescribed.

38 The applications A colored Tverberg type theorem with smaller number of colors, allowing k vertices of the same color in each face. We could consider topological Tverberg theorem and require the dimensions of faces of a simplex whose images intersect to be prescribed. Van Kampen-Flores theorem is an example of the result of this type.

39 The applications A colored Tverberg type theorem with smaller number of colors, allowing k vertices of the same color in each face. We could consider topological Tverberg theorem and require the dimensions of faces of a simplex whose images intersect to be prescribed. Van Kampen-Flores theorem is an example of the result of this type. We have to allow more points, i.e. to consider the mapping from the higher dimensional simplex, to be able to prescribe the dimensions of faces.

40 d-tverberg prescribable r-tuple [BFZ] P. Blagojević, F. Frick, G. Ziegler. Tverberg plus constraints. Bull. London Math. Soc. 46 (2014),

41 d-tverberg prescribable r-tuple [BFZ] P. Blagojević, F. Frick, G. Ziegler. Tverberg plus constraints. Bull. London Math. Soc. 46 (2014), Definition: For d 1 and r 2 an r-tuple (d 1, d 2,..., d r ) is d-tverberg prescribable if there is N such that for every continuous f : N R d there are disjoint faces σ 1, σ 2,..., σ r such that dim σ i = d i and f (σ 1 ) f (σ 2 ) f (σ r ).

42 d-tverberg prescribable r-tuple [BFZ] P. Blagojević, F. Frick, G. Ziegler. Tverberg plus constraints. Bull. London Math. Soc. 46 (2014), Definition: For d 1 and r 2 an r-tuple (d 1, d 2,..., d r ) is d-tverberg prescribable if there is N such that for every continuous f : N R d there are disjoint faces σ 1, σ 2,..., σ r such that dim σ i = d i and f (σ 1 ) f (σ 2 ) f (σ r ). If we don t require the dimensions of faces to be prescribed, such faces exist for N (r 1)(d + 1) by the Tverberg theorem.

43 d-tverberg prescribable r-tuple By an affine general position mapping we see that a d-tverberg prescribable r-tuple (d 1, d 2,..., d r ) has to satisfy d d r (r 1)d.

44 d-tverberg prescribable r-tuple By an affine general position mapping we see that a d-tverberg prescribable r-tuple (d 1, d 2,..., d r ) has to satisfy d d r (r 1)d. By an affine map mapping the vertices to N + 1 different points on the moment curve, we obtain a cyclic polytope as the image, and this forces d i [ d 2 ] to hold for every i.

45 d-admissible r-tuple Definition: For d 1 and r 2 an r-tuple (d 1, d 2,..., d r ) is d-admissible if [ ] d 2 di d for all i and d 1 + d d r (r 1)d.

46 d-admissible r-tuple Definition: For d 1 and r 2 an r-tuple (d 1, d 2,..., d r ) is d-admissible if [ ] d 2 di d for all i and d 1 + d d r (r 1)d. By previous remarks, we see that a d-tverberg prescribable r-tuple has to be d-admissible.

47 d-admissible r-tuple Definition: For d 1 and r 2 an r-tuple (d 1, d 2,..., d r ) is d-admissible if [ ] d 2 di d for all i and d 1 + d d r (r 1)d. By previous remarks, we see that a d-tverberg prescribable r-tuple has to be d-admissible. Question ([BFZ]): Is every d-admissible r-tuple d-tverberg prescribable?

48 d-admissible r-tuple Definition: For d 1 and r 2 an r-tuple (d 1, d 2,..., d r ) is d-admissible if [ ] d 2 di d for all i and d 1 + d d r (r 1)d. By previous remarks, we see that a d-tverberg prescribable r-tuple has to be d-admissible. Question ([BFZ]): Is every d-admissible r-tuple d-tverberg prescribable? Theorem ([BFZ]): Yes, if d 1 = d 2 = = d r.

49 d-admissible r-tuple Definition: For d 1 and r 2 an r-tuple (d 1, d 2,..., d r ) is d-admissible if [ ] d 2 di d for all i and d 1 + d d r (r 1)d. By previous remarks, we see that a d-tverberg prescribable r-tuple has to be d-admissible. Question ([BFZ]): Is every d-admissible r-tuple d-tverberg prescribable? Theorem ([BFZ]): Yes, if d 1 = d 2 = = d r. This follows also directly from the connectivity of multiple chessboard complexes.

50 A conjecture Conjecture ([BFZ]): Yes, if d 1 = = d s = d + 1 and d s+1 = = d r = d for some s.

51 A conjecture Conjecture ([BFZ]): Yes, if d 1 = = d s = d + 1 and d s+1 = = d r = d for some s. Conjecture ([BFZ]): Let r 2 be a prime power, d 1, N (r 1)(d + 2), and r(k + 1) + s > N + 1 for integers k 0 and 0 s < r. Then, for every continuous map f : N R d, there are r pairwise disjoint faces σ 1, σ 2,..., σ r of N such that f (σ 1 ) f (σ 2 ) f (σ r ), with dim σ i k + 1 for 1 i s and dim σ i k for s < i r.

52 A conjecture Conjecture ([BFZ]): Yes, if d 1 = = d s = d + 1 and d s+1 = = d r = d for some s. Conjecture ([BFZ]): Let r 2 be a prime power, d 1, N (r 1)(d + 2), and r(k + 1) + s > N + 1 for integers k 0 and 0 s < r. Then, for every continuous map f : N R d, there are r pairwise disjoint faces σ 1, σ 2,..., σ r of N such that f (σ 1 ) f (σ 2 ) f (σ r ), with dim σ i k + 1 for 1 i s and dim σ i k for s < i r. For a r-tuple (d 1, d 2,..., d r ), the configuration complex of all candidates is K = d 1+1,...,d r +1;1 N+1,r.

53 Symmetric multiple chessboard complexes There is an action of a symmetric group S r on this complex only if d 1 = d 2 = = d r.

54 Symmetric multiple chessboard complexes There is an action of a symmetric group S r on this complex only if d 1 = d 2 = = d r. We consider the symmetrized multiple chessboard complexes Σ k 1,...,k r ;1 m,r = S r k 1,...,k r ;1 m,r = kσ(1),...,kσ(r);1 m,r. σ S r

55 Symmetric multiple chessboard complexes There is an action of a symmetric group S r on this complex only if d 1 = d 2 = = d r. We consider the symmetrized multiple chessboard complexes Σ k 1,...,k r ;1 m,r = S r k 1,...,k r ;1 m,r = kσ(1),...,kσ(r);1 m,r. σ S r If r = p α is a prime power, there is a fixed point free action of the group (Z/p) α on the complex Σ k 1,...,k r ;1 m,r.

56 Topological properties Theorem. If m k k r + r 1, and max k i min k i = 1, the complex Σ k 1,...,k r ;1 m,r is shellable.

57 Topological properties Theorem. If m k k r + r 1, and max k i min k i = 1, the complex Σ k 1,...,k r ;1 m,r is shellable. Corollary. Under the same conditions, this complex is (k k r 2)-connected, and so there is no equivariant mapping from this complex to the appropriate representation sphere if r is a prime power.

58 Topological properties Theorem. If m k k r + r 1, and max k i min k i = 1, the complex Σ k 1,...,k r ;1 m,r is shellable. Corollary. Under the same conditions, this complex is (k k r 2)-connected, and so there is no equivariant mapping from this complex to the appropriate representation sphere if r is a prime power. An alternative proof of this consequence could be also provided using the established connectivity of k 1,...,k r ;1 m,r, and the Sarkaria inequality.

59 A conjecture is true Using this result, we are able to confirm the stronger version of the conjecture, obtained by replacing the original condition r(k + 1) + s > N + 1 by a weaker and more natural condition rk + s (r 1)d.

60 A conjecture is true Using this result, we are able to confirm the stronger version of the conjecture, obtained by replacing the original condition r(k + 1) + s > N + 1 by a weaker and more natural condition rk + s (r 1)d. This weaker condition is actually contained in the assumption that given r-tuple is d-admissible.

61 A conjecture is true Using this result, we are able to confirm the stronger version of the conjecture, obtained by replacing the original condition r(k + 1) + s > N + 1 by a weaker and more natural condition rk + s (r 1)d. This weaker condition is actually contained in the assumption that given r-tuple is d-admissible. Theorem: Every d-admissible r-tuple (d 1,..., d r ) satisfying max d i min d i = 1 is d-tverberg prescribable.

62 Preprints Duško Jojić, Siniša Vrećica, Rade Živaljević. Multiple chessboard complexes and the colored Tverberg problem. arxiv: Duško Jojić, Siniša Vrećica, Rade Živaljević. Symmetric multiple chessboard complexes and a new theorem of Tverberg type. arxiv:

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