Exact sequences for the homology of the matching complex

Building on work by Bouc and by Shareshian and Wachs, we provide a toolbox of long exact sequences for the reduced simplicial homology of the matching complex Mn, which is the simplicial complex of matchings in the complete graph Kn. Combining these sequences in different ways, we prove several results about the 3-torsion part of the homology of Mn. First, we demonstrate that there is nonvanishing 3-torsion in H̃d(Mn;Z) whenever νn ≤ d ≤ bn−6 2 c, where νn = d n−4 3 e. By results due to Bouc and to Shareshian and Wachs, H̃νn(Mn;Z) is a nontrivial elementary 3-group for almost all n and the bottom nonvanishing homology group of Mn for all n 6= 2. Second, we prove that H̃d(Mn;Z) is a nontrivial 3-group whenever νn ≤ d ≤ b2n−9 5 c. Third, for each k ≥ 0, we show that there is a polynomial fk(r) of degree 3k such that the dimension of H̃k−1+r(M2k+1+3r;Z3), viewed as a vector space over Z3, is at most fk(r) for all r ≥ k + 2.