Twisted actions on cohomologies and bimodules

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.