Eilenberg-MacLane spectra as equivariant Thom spectra

TL;DR

This paper demonstrates that certain equivariant Eilenberg-MacLane spectra can be realized as Thom spectra, extending previous results and providing new constructions and insights into equivariant stable homotopy theory.

Contribution

It generalizes the realization of equivariant Eilenberg-MacLane spectra as Thom spectra to all finite p-group actions and introduces new constructions and constraints relevant to equivariant spectra.

Findings

Equivariant mod p Eilenberg-MacLane spectra are Thom spectra for finite p-groups.

Constructs of alHalZ_{(p)} are established.

Constraints on Hurewicz images of equivariant spectra with norms are identified.

Abstract

We prove that the $G$-equivariant mod $p$ Eilenberg--MacLane spectrum arises as an equivariant Thom spectrum for any finite, $p$-power cyclic group $G$, generalizing a result of Behrens and the second author in the case of the group $C_2$. We also establish a construction of $\mathrm{H}\underline{\mathbb{Z}}_{(p)}$, and prove intermediate results that may be of independent interest. Highlights include constraints on the Hurewicz images of equivariant spectra that admit norms, and an analysis of the extent to which the non-equivariant $\mathrm{H}\mathbb{F}_p$ arises as the Thom spectrum of a more than double loop map.