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

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 $2$ and 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 $H\backslash G/K$. 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 $\mathbf{B}(A^{\prime },A,A")$ is defined, valid when $A^{\prime }\leftarrow A\to A"$ is a pair of maps of homotopy Gerstenhaber algebras.

34-page version to be published in Alg. Geom. Top. Appendix joint with Matthias Franz. This version differs from arXiv v1 in having a few typos corrected, a new final section with a sample computation added, and much background and detail removed, including some proofs that otherwise are unpublished

References in corpus (2)

• Homotopy Gerstenhaber algebras are strongly homotopy commutative
• Szczarba's twisting cochain is comultiplicative

Cited by in corpus (1)

• E2 formality via obstruction theory