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., 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, First, we demonstrate that there is nonvanishing 3-torsion in (H) over bar (d)(M-n : Z) whenever v(n) <= d <= [n-6/2], where v(n) =[n-4/3]. By results due to Bouc and to Shareshian and Wachs, (H) over bar (d)(M-n : Z) is a nontrivial elementary 3-group for almost all n and the bottom nonvanishing homology group of M. for all n 0 2. Second, we prove that (H) over bar (d)(M-n : Z) is a nontrivial 3-group whenever v(n) <= d <= [2n-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) over bar (k-1+r) (M2k+1+3r:Z(3)), viewed as a vector space over Z(3), is at most f(k)(r) for all r >= k + 2.

Nyckelord

Matching complex

Simplicial homology

Long exact sequence

chessboard complexes

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