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The equivariant cohomology for semidirect product actions

Sergio Chaves

Abstract

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 $BG$ for field coefficients. We study the particular case when $G$ is a semi-direct product and $K$ is its maximal elementary abelian 2-subgroup for cohomology with coefficients in a field of characteristic two. This provides a different approach to investigate the syzygy order of 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.

The equivariant cohomology for semidirect product actions

Abstract

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 when there is a closed subgroup such that the cohomology of the classifying space is free over the cohomology of for field coefficients. We study the particular case when is a semi-direct product and is its maximal elementary abelian 2-subgroup for cohomology with coefficients in a field of characteristic two. This provides a different approach to investigate the syzygy order of 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.

Paper Structure

This paper contains 4 sections, 18 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Free extension pairs
  3. Torus actions and compatible involutions
  4. Reduction to 2-torus actions

Key Result

Theorem 1.1

Let $G$ be a compact Lie group that admits a closed subgroup $K$ satisfying the conditions above. Let $X$ be a $G$-space such that $H^*(X)^G = H^*(X)$. Then $W$ acts on the $K$-equivariant cohomology of $X$, there is a natural isomorphism of $H^*(BG)$-algebras $H^*_G(X) \cong H^*_K(X)^W$ and a natur

Theorems & Definitions (36)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Example 2.4
  • Proposition 2.5
  • proof
  • ...and 26 more