Twisted actions on cohomologies and bimodules

TL;DR

This paper introduces a twisted action of equivariant cohomology on the cohomology of L-spaces, establishing a bimodule structure that leads to an equivariant Künneth isomorphism and applications to Bott-Samelson varieties.

Contribution

It presents a novel twisted action and bimodule framework for equivariant cohomology, enabling new computations and morphisms for Bott-Samelson varieties.

Findings

Established a bimodule structure on equivariant cohomology

Proved an equivariant Künneth isomorphism

Computed cohomologies of Bott-Samelson varieties

Abstract

We introduce a twisted action of the equivariant cohomology of the singleton $H_R^\bullet({\rm pt},\Bbbk)$ on the equivarinat cohomology $H_L^\bullet(X,\Bbbk)$ of an $L$-space $X$. Considering this actions as a right action, $H_L^\bullet(X,\Bbbk)$ becomes a bimodule togeather with the canonical left action of $H_L^\bullet({\rm pt},\Bbbk)$. Using this bimodule structure, we prove an equivariant version of the K\"unneth isomorphism. We apply this result to the computation of the equivariant cohomologies of Bott-Samelson varieties and to a geometric construction of the bimodule morphisms between these cohomologies.