TOP HOMOLOGY OF HYPERGRAPH MATCHING COMPLEXES, p-CYCLE COMPLEXES AND QUILLEN COMPLEXES OF SYMMETRIC GROUPS

We investigate the representation of a symmetric group Sn 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 Sn 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.