Symmetric Powers and the Equivariant Dual Steenrod Algebra

Abstract

The structure of the Steenrod algebra of stable mod p cohomology operations and its dual A_* was worked out completely by Milnor - for every prime p, A_* is a graded-commutative Hopf algebra. However, much of this structure can be alternatively found using the filtration of HZ coming from the symmetric powers of the sphere spectrum, first studied by Nakaoka, and later by Mitchell-Priddy and others. This filtration not only realizes elements of A_* explicitly in the homology of certain spaces, but also is the object of the Whitehead Conjecture (proven by Kuhn) and satisfies a duality with the Goodwillie tower of a sphere. Inspired by the approach of Mitchell-Priddy, we use an equivariant analogue of the symmetric power filtration to try to compute a similar algebra decomposition of HF_p \sm HF_p in the category of HF_p-modules, where F_p is the constant Mackey functor. We focus on the case where G=C_p, the cyclic group of order p. In particular, we show that the cofibers in the symmetric power filtration of HF_p are Steinberg summands of equivariant classifying spaces, and these cofibers stably split after smashing with HF_p. We also explicitly compute equivariant homology decompositions of these spaces when p=2.