Twisted actions on cohomologies and bimodules

We introduce a twisted action of the equivariant cohomology of the singleton $H_R^{\bullet }(\mathrm{p}\mathrm{t},\mathbb{k})$ on the equivarinat cohomology $H_L^{\bullet }(X,\mathbb{k})$ of an $L$-space $X$. Considering this actions as a right action, $H_L^{\bullet }(X,\mathbb{k})$ becomes a bimodule togeather with the canonical left action of $H_L^{\bullet }(\mathrm{p}\mathrm{t},\mathbb{k})$. Using this bimodule structure, we prove an equivariant version of the Künneth 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.