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

摘要

We investigate the representation of a symmetric group Sn onthe homology of its Quillen complex at a prime p. For homol-ogy groups in small codimension, we derive an explicit formulafor this representation in terms of the representations of symmet-ric groups on homology groups of p-uniform hypergraph matchingcomplexes. We conjecture an explicit formula for the representa-tion of Sn on the top homology group of the corresponding hy-pergraph matching complex when n≡1 mod p. Our conjecturefollows from work of Bouc when p = 2, and we prove the conjec-ture when p = 3.