Equivariant cohomology and depth

TL;DR

This paper investigates the algebraic structure of equivariant cohomology modules for finite V-CW-complexes, establishing that the depth of these modules decreases or remains stable under subgroup restrictions.

Contribution

It proves that the depth of equivariant cohomology modules over subgroup actions is always less than or equal to that over the whole group, revealing a monotonicity property.

Findings

Depth of equivariant cohomology modules is non-increasing under subgroup restriction.

The result applies to finite V-CW-complexes with mod 2 cohomology.

Provides a new inequality relating subgroup and group equivariant cohomology depths.

Abstract

Let $n \geq 1$ be an integer, let $V=(\mathbb{Z}/2\mathbb{Z})^{n}$ and let $X$ be a $V$-CW-complex. If $X$ is a finite $CW$-complexe, the equivariant modulo $2$ cohomology of the $V$-CW-complexe $X$, denoted by $H_{V}^{*}(X, \mathbb{F}_{2})$, is a finite type module over the modulo $2$ cohomology of the group $V$, denoted by $H^{*}(V, \mathbb{F}_{2})$. Let $dth_{H^{*}V}H_{V}^{*}(X, \mathbb{F}_{2})$ be the depth of the finite type $H^{*}(V, \mathbb{F}_{2})$-module $H_{V}^{*}(X, \mathbb{F}_{2})$ relatively to the augmentation ideal, $\widetilde{H^{*}}(V, \mathbb{F}_{2})$, of $H^*(V, \mathbb{F}_{2})$. \medskip\\ The aim of this paper is to prove the following result: \medskip\\ {\bf Theorem}: For every subgroup $W$ of $V$, we have: $dth_{H^{*}W}H_{W}^{*}(X, \mathbb{F}_{2}) \leq dth_{H^{*}V} H_{V}^{*}(X, \mathbb{F}_{2}) $.