Symmetric Powers and Eilenberg--Maclane Spectra

Abstract

We filter the equivariant Eilenberg Maclane spectrum HF p using the mod p symmetric powers of the equivariant sphere spectrum, Sp ∞ Z/p (Σ ∞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 (Z/p) n for n = 0, 1, 2, . . .. We show that the layers of the filtration split after smashing with HF p . Along the way, we produced a general computation of the geometric fixed points of HZ and HF p by using symmetric powers.