The cohomology of biquotients via a product on the two-sided bar construction

Carlson, Jeffrey D

doi:10.2140/agt.2025.25.5279

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 cohomology of biquotients via a product on the two-sided bar construction

Jeffrey D Carlson

Appendix: Jeffrey D Carlson and Matthias Franz

Algebraic & Geometric Topology 25 (2025) 5279–5317

DOI: 10.2140/agt.2025.25.5279

Abstract

We compute the Borel equivariant cohomology ring of the left $K$ -action on a homogeneous space $G ∕ H$ , where $G$ is a connected Lie group, $H$ and $K$ are closed connected subgroups, and $2$ as well as the torsion primes of the Lie groups are units of the coefficient ring. As a special case, this gives the singular cohomology rings of biquotients $H ∖ G ∕ K$ . This depends on a version of the Eilenberg–Moore theorem developed in the appendix, where a novel multiplicative structure on the two-sided bar construction $B (A^{''}, A, A^{'})$ is defined, valid when $A^{''} \leftarrow A \to A$ is a pair of maps of homotopy Gerstenhaber algebras.

Keywords

cohomology, equivariant cohomology, homogeneous spaces, biquotients, homotopy Gerstenhaber algebras, strongly homotopy-commutative algebras, SHC-algebras, $A$-infinity algebras, Eilenberg–Moore, bar construction, two-sided bar construction, product

Mathematical Subject Classification

Primary: 16U80, 55N91, 57T15, 57T30

Secondary: 16E45, 55T20, 57T35

References

Publication

Received: 25 February 2022

Revised: 13 March 2023

Accepted: 11 April 2023

Published: 18 December 2025

Authors

Jeffrey D Carlson
Department of Mathematics Tufts University Medford, MA United States

Jeffrey D Carlson


Matthias Franz
Department of Mathematics University of Western Ontario London ON Canada

Open Access made possible by participating institutions via Subscribe to Open.

Volume 25, issue 9 (2025)