Mathematics – Combinatorics
Scientific paper
2012-03-26
J. Combin. Theory Ser. A 115 (2008) 1504-1526
Mathematics
Combinatorics
31 pages
Scientific paper
10.1016/j.jcta.2008.03.001
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 $\nu_n \le d \le (n-6}/2$, where $\nu_n= \lceil (n-4)/3 \rceil$. By results due to Bouc and to Shareshian and Wachs, $H_{\nu_n}(M_n;Z)$ is a nontrivial elementary 3-group for almost all $n$ and the bottom nonvanishing homology group of $M_n$ for all $n \neq 2$. Second, we prove that $H_d(M_n;Z)$ is a nontrivial 3-group whenever $\nu_n \le d \le (2n-9)/5$. Third, for each $k \ge 0$, we show that there is a polynomial $f_k(r)$ of degree 3k such that the dimension of $H_{k-1+r}(M_{2k+1+3r};Z_3)$, viewed as a vector space over $Z_3$, is at most $f_k(r)$ for all $r \ge k+2$.
No associations
LandOfFree
Exact Sequences for the Homology of the Matching Complex does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Exact Sequences for the Homology of the Matching Complex, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Exact Sequences for the Homology of the Matching Complex will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-641883