Steinberg Summands in the free ${\mathbb{F}}_p$-module on the Equivariant Sphere Spectrum

Abstract

Let $G$ be a finite $p$-group. The Eilenberg-Maclane spectrum of the constant Mackey functor ${\underline{\mathbb{F}}}_p$, denoted $H{\underline{\mathbb{F}}}_p$, is modeled by the free ${\mathbb{F}}_p$-module on the $G$-equivariant sphere spectrum. With this construction, one has a `word length' filtration $\{(H{\underline{\mathbb{F}}}_p{)}_n{\}}_{n\ge 1}$. Our main theorem is that the $k$-th layer $(H{\underline{\mathbb{F}}}_p{)}_{p^k}/(H{\underline{\mathbb{F}}}_p{)}_{p^{k-1}}$ is $p$-locally equivalent to the $k$-fold suspension of the Steinberg summand of the $G$-equivariant classifying space of $(\mathbb{Z}/p{)}^k$. This is a generalization of the main result of [21]. We also show that when one smashes this filtration with $H{\underline{\mathbb{F}}}_p$, the filtration splits into its layers. The future goal of this work is to compute the $C_p$-equivariant dual Steenrod algebra $H{\underline{\mathbb{F}}}_p\wedge H{\underline{\mathbb{F}}}_p$ when $p>2$, via explicit cellular constructions of equivariant classifying spaces.

The paper has been subsumed by a new paper, "Symmetric Powers and Eilenberg-Maclane Spectra" [arXiv:1904.01708], which strengthens the results, fixes some errors, and significantly improves the organization and narrative