Inessential Brown-Peterson homology and bordism of elementary abelian groups

Bernhard Hanke

We compute the equivariant bordism of free oriented (Z/p)n-manifolds as a module over ΩSO*, when p is an odd prime. We show, among others, that this module is canonically isomorphic to a direct sum of suspensions of multiple tensor products of ΩSO*(BZ/p), and that it is generated by products of standard lens spaces. This considerably improves previous calculations by various authors.

Our approach relies on the investigation of the submodule of the Brown-Peterson homology of B(Z/p)n generated by elements coming from proper subgroups of (Z/p)n.

We apply our results to the Gromov-Lawson-Rosenberg conjecture for atoral manifolds whose fundamental groups are elementary abelian of odd order.