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 M n , which is the simplicial complex of matchings in the complete graph K n . Combining these sequences in different ways, we prove several results about the 3-torsion part of the homology of M n . First, we demonstrate that there is nonvanishing 3-torsion in H ˜ d ( M n ; Z ) whenever ν n ⩽ d ⩽ ⌊ n − 6 2 ⌋ , where ν n = ⌈ n − 4 3 ⌉ . By results due to Bouc and to Shareshian and Wachs, H ˜ ν n ( M n ; Z ) is a nontrivial elementary 3-group for almost all n and the bottom nonvanishing homology group of M n for all n ≠ 2 . Second, we prove that H ˜ d ( M n ; Z ) is a nontrivial 3-group whenever ν n ⩽ d ⩽ ⌊ 2 n − 9 5 ⌋ . Third, for each k ⩾ 0 , we show that there is a polynomial f k ( r ) of degree 3k such that the dimension of H ˜ k − 1 + r ( M 2 k + 1 + 3 r ; Z 3 ) , viewed as a vector space over Z 3 , is at most f k ( r ) for all r ⩾ k + 2 .