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

Let $1 \le m \le n$. 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 $H_d(M_{m,n};Z)$ whenever $\frac{m+n-4}{3} \le d \le m-4$ and whenever $6 \le m < n$ 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 $H_{d}(M_{m,n};Z) \neq 0$. Second, for each $k \ge 0$, we show that there is a polynomial $f_k(a,b)$ of degree 3k such that the dimension of $H_{k+a+2b-2}(M_{k+a+3b-1,k+2a+3b-1};Z_3)$, viewed as a vector space over $Z_3$, is at most $f_k(a,b)$ for all $a \ge 0$ and $b \ge k+2$. Third, we give a computer-free proof that $H_2(M_{5,5};Z) \cong Z_3$. 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 $M_{m-2,n-3}$.