The Borel cohomology of free iterated loop spaces

Levy, Ishan; Wu, Justin

doi:10.2140/agt.2026.26.1

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

Submission form

Editorial board

Subscriptions

ISSN (electronic): 1472-2739

ISSN (print): 1472-2747

Author index

To appear

Other MSP journals

The Borel cohomology of free iterated loop spaces

Ishan Levy and Justin Wu

Algebraic & Geometric Topology 26 (2026) 1–27

DOI: 10.2140/agt.2026.26.1

Abstract

We compute the $SO (n + 1)$ -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 Bökstedt and Ottosen’s computation for the free loop space. The highlight of our computation is a construction of cohomology classes using an $O (n)$ -equivariant evaluation map and a pushforward map.

Keywords

equivariant cohomology, free loop space, Steenrod algebra

Mathematical Subject Classification

Primary: 55N91, 55S10

References

Publication

Received: 12 September 2021

Revised: 26 January 2025

Accepted: 17 February 2025

Published: 16 January 2026

Authors

Ishan Levy
School of Mathematics Institute for Advanced Study Princeton, NJ United States

Justin Wu
Department of Mathematics University of California, Berkeley Berkeley, CA United States

This article is currently available only to readers at paying institutions. If enough institutions subscribe to this Subscribe to Open journal for 2026, the article will become Open Access in early 2026. Otherwise, this article (and all 2026 articles) will be available only to paid subscribers.

Volume 26, issue 1 (2026)