Top homology of hypergraph matching complexes, p-cycle complexes and Quillen complexes of symmetric groups

Abstract

We investigate the representation of a symmetric group S n on the homology of its Quillen complex at a prime p. For homology groups in small codimension, we derive an explicit formula for this representation in terms of the representations of symmetric groups on homology groups of p-uniform hypergraph matching complexes. We conjecture an explicit formula for the representation of S n on the top homology group of the corresponding hypergraph matching complex when n ≡ 1 mod p. Our conjecture follows from work of Bouc when p = 2, and we prove the conjecture when p = 3.

On a conjecture of Stein

“…Theorem 3.11. η H (M(K n,n )) ⌊ 2n 3 ⌋. In [17] it was shown that in fact equality holds in Theorem 3.11.…”

Poset topology: Tools and applications

“…He used simple connectivity of M n for n ≥ 8, which was proved by Bouc, to establish simple connectivity of S 2 (S n ) for n ≥ 8. It was also shown by Ksontini [114,115,116], Shareshian [154], and Shareshian and Wachs [156] that a hypergraph version of the matching complex is useful in studying the topology of ∆(S p (S n )) when p ≥ 3. The next two examples are connected with the combinatorics of knot spaces and arose in the work of Vassiliev [191,192,193].…”

“…(b) Show that each f −1 (P (∆) ≤F ) is a wedge of ν(F, m) spheres of dimension dim F . Shareshian [154] and Shareshian and Wachs [156] use Theorems 5.5.2 and 5.5.4, to derive information about the homology of the Quillen complex A p (S n ) from a hypergraph version of the matching complex. Another application of Theorems 5.5.2 and 5.5.4 can be found in [131].…”

Matching and Independence Complexes Related to Small Grids

“…However, matching and independence complexes quickly become quite complicated, e.g. [2,3,4,6,7,8,13,16,17,18]. Jonsson [12] provides a thorough survey regarding these and other simplicial complexes arising from graphs, with special emphasis on the matching complex for complete graphs and complete bipartite graphs.…”