Twisted Actions on Cohomologies and Bimodules

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.