Truncated Quillen coplexes of p-groups

Abstract

Let p be an odd prime and let P be a p-group. We examine the order complex of the poset of elementary abelian subgroups of P having order at least p^2. S. Bouc and J. Th\'evenaz showed that this complex has the homotopy type of a wedge of spheres. We show that, for each nonnegative integer l, the number of spheres of dimension l in this wedge is controlled by the number of extraspecial subgroups X of P having order p^{2l+3} and satisfying Omega_1(C_P(X))=Z(X). We go on to provide a negative answer to a question raised by Bouc and Th\'evenaz concerning restrictions on the homology groups of the given complex.