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

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 (H)\tilde]_d(\sf M_m,n; \mathbb Z){{\tilde{H}_d({\sf M}_{m,n}; {\mathbb Z})}} whenever \fracm+n-43 ￡ d ￡ m-4{{\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 (H)\tilde]_d (\sf M_m,n; \mathbb Z) 1 0{{\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 (H)\tilde]_k+a+2b-2 (\sf M_{k+a+3b-1,k+2a+3b-1}; \mathbb Z₃){{\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 \mathbbZ₃{\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 (H)\tilde]₂ (\sf M_5,5; \mathbb Z) @ \mathbb Z₃{{\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.