Torsion in the matching complex and chessboard complex

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. Björner, Lovász, Vrećica and Živaljević established bounds on the...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2007-07, Vol.212 (2), p.525-570
Main Authors: Shareshian, John, Wachs, Michelle L.
Format: Article
Language:English
Subjects:
Chessboard complex
Homology
Matching complex
Torsion
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Summary: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. Björner, Lovász, Vrećica and Živaljević 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.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2006.10.014