Twisted Actions on Cohomologies and Bimodules

Abstract For closed subgroups L and R of a compact Lie group G, a left L -space X, and an L -equivariant continuous map A: X G/R, we introduce the twisted action of the equivariant cohomology HR^ (pt, ) on the equivariant cohomology HL^ (X, ). Considering this action as a right action, HL^ (X, ) becomes a bimodule together with the canonical left action of HL^ (pt, ). 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 them.