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TWISTED ACTIONS ON COHOMOLOGIES AND BIMODULES
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 117 / Issue 2 / October 2024
- Published online by Cambridge University Press:
- 31 May 2024, pp. 187-238
- Print publication:
- October 2024
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- Article
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For closed subgroups L and R of a compact Lie group G, a left L-space X, and an L-equivariant continuous map
$A:X\to G/R$, we introduce the twisted action of the equivariant cohomology 
$H_R^{\bullet }(\mathrm {pt},\Bbbk )$ on the equivariant cohomology 
$H_L^{\bullet }(X,\Bbbk )$. Considering this action as a right action, 
$H_L^{\bullet }(X,\Bbbk )$ becomes a bimodule together with the canonical left action of 
$H_L^{\bullet }(\mathrm {pt},\Bbbk )$. 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.
