Torsion in the matching complex and chessboard complex

Abstract Topological properties of the matching complex were first studied by Bouc in connection with Quillen complexes, and topological properties of the chessboard complex were first studied by Garst in connection with Tits coset complexes. Bjorner, Lovasz, Vrecica and Živaljevic established bounds on the connectivity of these complexes and conjectured that these bounds are sharp. In this paper we show that the conjecture is true by establishing the nonvanishing of integral homology in the degrees given by these bounds. Moreover, we show that for sufficiently large n, the bottom nonvanishing homology of the matching complex M n is an elementary 3-group, improving a result of Bouc, and that the bottom nonvanishing homology of the chessboard complex M n , n is a 3-group of exponent at most 9. When n ≡ 2 mod 3 , the bottom nonvanishing homology of M n , n is shown to be Z 3 . Our proofs rely on computer calculations, long exact sequences, representation theory, and tableau combinatorics.

[1]  James R. Munkres,et al.  Elements of algebraic topology , 1984 .

[2]  Michelle L. Wachs,et al.  Shellable nonpure complexes and posets. II , 1996 .

[3]  M. Wachs.  Topology of matching, chessboard, and general bounded degree graph complexes , 2003 .

[4]  P. Garst,et al.  Cohen-macaulay complexes and group actions. , 1979 .

[5]  Bruce E. Sagan,et al.  The symmetric group - representations, combinatorial algorithms, and symmetric functions , 2001, Wadsworth & Brooks / Cole mathematics series.

[6]  Sinisa T. Vrecica,et al.  The Colored Tverberg's Problem and Complexes of Injective Functions , 1992, J. Comb. Theory A.

[7]  Joel Friedman,et al.  On the Betti Numbers of Chessboard Complexes , 1998 .

[8]  W. Fulton.  Young Tableaux: With Applications to Representation Theory and Geometry , 1996 .

[9]  László Lovász,et al.  Chessboard Complexes and Matching Complexes , 1994 .

[10]  Volkmar Welker,et al.  Complexes of Directed Graphs , 1999, SIAM J. Discret. Math..

[11]  M. Wachs.  SHELLABLE NONPURE COMPLEXES AND POSETS. I , 1996 .

[12]  Christos A. Athanasiadis.  Decompositions and Connectivity of Matching and Chessboard Complexes , 2004, Discret. Comput. Geom..

[13]  R. Forman.  Morse Theory for Cell Complexes , 1998 .

[14]  Svante Linusson,et al.  COMPLEXES OF NOT i-CONNECTED GRAPHS , 1997, math/9705219.

[15]  S. Bouc,et al.  Homologie de certains ensembles de 2-sous-groupes des groupes symétriques , 1992 .

[16]  Günter M. Ziegler,et al.  Shellability of chessboard complexes , 1994 .