Steinberg Summands and Symmetric Powers of the G-Sphere

Krishanu Roy Sankar

Published 2017-11-15Version 1

Let $G$ be a finite abelian $p$-group. We use the symmetric powers of the $G$-equivariant sphere spectrum to obtain a filtration for $H\underline{\mathbb{F}}_p$, the Eilenberg-Maclane spectrum for the constant Mackey functor $\underline{\mathbb{F}}_p$. Our main theorem is that there is an equivalence between the $k$-th cofiber of this filtration and the Steinberg summand of the $G$-equivariant classifying space of $(\mathbb{Z}/p)^k$. We also show that when one smashes with $H\underline{\mathbb{F}}_p$, the filtration splits into its associated graded. In a future paper, we will use this result to compute the equivariant dual Steenrod algebra $H\underline{\mathbb{F}}_p\wedge H\underline{\mathbb{F}}_p$ at odd primes via explicit cellular constructions of these equivariant classifying spaces.