The Borel Cohomology of Free Iterated Loop Spaces

TL;DR

This paper computes the equivariant mod 2 Borel cohomology of free iterated loop spaces for certain spaces, extending known results for free loop spaces and introducing new cohomology class constructions.

Contribution

It provides a new computation method for the Borel cohomology of free iterated loop spaces using equivariant evaluation and pushforward maps, generalizing previous results.

Findings

Explicit cohomology computations for free iterated loop spaces

Extension of known results for free loop spaces to higher iterates

Introduction of a new cohomology class construction method

Abstract

We compute the $\rm{SO}(n+1)$-equivariant mod $2$ Borel cohomology of the free iterated loop space $Z^{S^n}$ when $Z$ is a mod $2$ generalized Eilenberg Mac Lane space. When $n=1$, this recovers B\"okstedt and Ottosen's computation for the free loop space. The highlight of our computation is a construction of cohomology classes using an $\mathrm{O}(n)$-equivariant evaluation map and a pushforward map.