Torsion in the matching complex and chessboard complex
By John Shareshian, Michelle L. Wachs. 2004
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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. 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.
Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/b72128800724e199585c15e9b96ca192b0869c28
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In Computational Theory and Mathematics
In Discrete Mathematics and Combinatorics
In Mathematical Physics
Citations in the corpus
- Optimal bounds for the colored Tverberg problem (2009, background): “However, they loose a factor of 2 from the fact that the chessboard-complexes ∆r,t of dimension r − 1 are homologically (r − 2)-connected only if t ≥ 2r − 1; compare [5], [21], and [14].”
- Beyond the Borsuk–Ulam Theorem: The Topological Tverberg Story (2016, background): “Its connectivity lower bound is best possible according to [41].”