On the 3-Torsion Part of the Homology of the Chessboard Complex

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Let 1 ≤ m ≤ n. We prove various results about the chessboard complex Mm,n, which is the simplicial complex of matchings in the complete bipartite graph Km,n. First, we demonstrate that there is nonvanishing 3-torsion in \({{\tilde{H}_d({\sf M}_{m,n}; {\mathbb Z})}}\) whenever \({{\frac{m+n-4}{3}\leq d \leq m-4}}\) and whenever 6 ≤ m < n and d = m − 3. Combining this result with theorems due to Friedman and Hanlon and to Shareshian and Wachs, we characterize all triples (m, n, d ) satisfying \({{\tilde{H}_d \left({\sf M}_{m,n}; {\mathbb Z}\right) \neq 0}}\). Second, for each k ≥ 0, we show that there is a polynomial fk(a, b) of degree 3k such that the dimension of \({{\tilde{H}_{k+a+2b-2}}\,\left({{\sf M}_{k+a+3b-1,k+2a+3b-1}}; \mathbb Z_{3}\right)}\), viewed as a vector space over \({\mathbb{Z}_3}\), is at most fk(a, b) for all a ≥ 0 and b ≥ k + 2. Third, we give a computer-free proof that \({{\tilde{H}_2 ({\sf M}_{5,5}; \mathbb {Z})\cong \mathbb Z_{3}}}\). Several proofs are based on a new long exact sequence relating the homology of a certain subcomplex of Mm,n to the homology of Mm-2,n-1 and Mm-2,n-3.