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 Hd(Mn;Z) whenever
\nu_n \le d \le (n-6}/2, where νn=⌈(n−4)/3⌉. By results due
to Bouc and to Shareshian and Wachs, Hνn(Mn;Z) is a nontrivial
elementary 3-group for almost all n and the bottom nonvanishing homology
group of Mn for all n=2. Second, we prove that Hd(Mn;Z) is a
nontrivial 3-group whenever νn≤d≤(2n−9)/5. Third, for each k≥0, we show that there is a polynomial fk(r) of degree 3k such that the
dimension of Hk−1+r(M2k+1+3r;Z3), viewed as a vector space over Z3,
is at most fk(r) for all r≥k+2.Comment: 31 page