Symmetric Powers and Eilenberg--Maclane Spectra

Abstract

We filter the equivariant Eilenberg Maclane spectrum $H{\underline{\mathbb{F}}}_p$ using the mod $p$ symmetric powers of the equivariant sphere spectrum, ${\mathrm{Sp}}_{\mathbb{Z}/p}^{\infty }({\Sigma }^{\infty G}{S}^0)$. When $G$ is a $p$-group, we show that the layers in the filtration are the Steinberg summands of the equivariant classifying spaces of $(\mathbb{Z}/p{)}^n$ for $n=0,1,2,\dots$. We show that the layers of the filtration split after smashing with $H{\underline{\mathbb{F}}}_p$. Along the way, we produced a general computation of the geometric fixed points of $H\underline{\mathbb{Z}}$ and $H{\underline{\mathbb{F}}}_p$ by using symmetric powers.

47 pages