Torsion in the matching complex and chessboard complex

John Shareshian, Michelle L. Wachs

Research output: Contribution to journalArticle

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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. 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 Mn is an elementary 3-group, improving a result of Bouc, and that the bottom nonvanishing homology of the chessboard complex Mn, n is a 3-group of exponent at most 9. When n ≡ 2 mod 3, the bottom nonvanishing homology of Mn, n is shown to be Z3. Our proofs rely on computer calculations, long exact sequences, representation theory, and tableau combinatorics.

Original languageEnglish (US)
Pages (from-to)525-570
Number of pages46
JournalAdvances in Mathematics
Volume212
Issue number2
DOIs
StatePublished - Jul 10 2007

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Keywords

  • Chessboard complex
  • Homology
  • Matching complex
  • Torsion

ASJC Scopus subject areas

  • Mathematics(all)

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