Cohomology rings of extended powers and free infinite loop spaces

TL;DR

This paper computes the mod-p cohomology of extended powers and their group completions, revealing a Hopf ring structure that simplifies cohomology calculations for free infinite loop spaces.

Contribution

It introduces a Hopf ring framework with divided powers for cohomology of extended powers, extending previous work on symmetric groups and infinite loop spaces.

Findings

Cohomology of extended powers is computed using a Hopf ring structure.

The framework simplifies calculations of cohomology rings of free infinite loop spaces.

New results on symmetric groups with sign representation are incorporated.

Abstract

We calculate mod-p cohomology of extended powers, and their group completions which are free infinite loop spaces. We consider the cohomology of all extended powers of a space together and identify a Hopf ring structure with divided powers within which cup product structure is more readily computable than on its own. We build on our previous calculations of cohomology of symmetric groups, which are the cohomology of extended powers of a point, the well-known calculation of homology, and new results on cohomology of symmetric groups with coefficients in the sign representation. We then use this framework to understand cohomology rings of related spaces such as infinite extended powers and free infinite loop spaces. v2 typo in gradings of Theorem 2.35 corrected.