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Let 1 ≤ m ≤ n. We prove various results about the chessboard complex M  m,n , which is the simplicial complex of matchings in the complete bipartite graph K  m,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 f  k (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 f  k (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 M  m,n  to the homology of M  m-2,n-1 and M  m-2,n-3.

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