More torsion in the homology of the matching complex.(English)Zbl 1247.05271

Summary: A matching on a set $$X$$ is a collection of pairwise disjoint subsets of $$X$$ of size two. Using computers, we analyze the integral homology of the matching complex $$M_n$$, which is the simplicial complex of matchings on the set $$\{1,\dots, n\}$$. The main result is the detection of elements of order $$p$$ in the homology for $$p \in \{5, 7, 11, 13\}$$. Specifically, we show that there are elements of order 5 in the homology of $$M_n$$ for $$n \geq 18$$ and for $$n \in \{14,16\}$$. The only previously known value was $$n = 14$$, and in this particular case we have a new computer-free proof. Moreover, we show that there are elements of order 7 in the homology of $$M_n$$ for all odd $$n$$ between $$23$$ and $$41$$ and for $$n = 30$$. In addition, there are elements of order $$11$$ in the homology of $$M_{47}$$ and elements of order $$13$$ in the homology of $$M_{62}$$. Finally, we compute the ranks of the Sylow 3- and 5-subgroups of the torsion part of $$\tilde{H}_d(M_n; \mathbb{Z})$$ for $$13 \leq n \leq 16$$; a complete description of the homology already exists for $$n \leq 12$$. To prove the results, we use a representation-theoretic approach, examining subcomplexes of the chain complex of $$M_n$$ obtained by letting certain groups act on the chain complex.

MSC:

 05E18 Group actions on combinatorial structures 55U10 Simplicial sets and complexes in algebraic topology
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References:

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