Bordism of elementary abelian groups via inessential Brown-Peterson homology

TL;DR

This paper computes the equivariant bordism of free oriented elementary abelian p-groups using inessential Brown-Peterson homology, revealing a structure as a sum of suspensions of tensor products and linking it to lens spaces.

Contribution

It introduces a new computation method for equivariant bordism of elementary abelian groups via inessential Brown-Peterson homology, improving previous results.

Findings

The bordism module is isomorphic to a sum of suspensions of tensor products of $oldsymbol{ ext{Ω}^{SO}_*(B ext{Z}/p)}$.

Generated by products of standard lens spaces.

Applications to the Gromov-Lawson-Rosenberg conjecture for atoral manifolds with elementary abelian fundamental groups.

Abstract

We compute the equivariant bordism of free oriented $(\mathbb{Z}/p)^n$-manifolds as a module over $\Omega_*^{SO}$, 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 $\Omega^{SO}_*(B \mathbb{Z}/p)$, 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 $B (\mathbb{Z}/p)^n$ generated by elements coming from proper subgroups of $(\mathbb{Z}/p)^n$. We apply our results to the Gromov-Lawson-Rosenberg conjecture for atoral manifolds whose fundamental groups are elementary abelian of odd order.