EQUIVARIANT COHOMOLOGY FOR SEMIDIRECT PRODUCT ACTIONS

The rational Borel equivariant cohomology for actions of a compact connected Lie group is determined by restriction of the action to a maximal torus. We show that a similar reduction holds for any compact Lie group G when there is a closed subgroup K such that the cohomology of the classifying space BK is free over the cohomology of BG with field coefficients. This provides a different approach to the equivariant cohomology of a space with a torus action and a compatible involution, and we relate this description with results for 2 torus actions.