Freeness of equivariant cohomology and mutants of compactified representations

TL;DR

This paper surveys generalizations of the Chang-Skjelbred Lemma for integral coefficients and constructs examples of manifolds with torus actions where equivariant cohomology is torsion-free but not free, answering a specific open question.

Contribution

It introduces new examples of manifolds with torsion-free but non-free equivariant cohomology, using compactified representations and Hopf bundles.

Findings

Constructed manifolds with torsion-free but non-free equivariant cohomology.

Provided a survey of generalizations of the Chang-Skjelbred Lemma.

Answered an open question by Allday about the freeness of equivariant cohomology.

Abstract

We survey generalisations of the Chang-Skjelbred Lemma for integral coefficients. Moreover, we construct examples of manifolds with actions of tori of rank > 2 whose equivariant cohomology is torsion-free, but not free. This answers a question of Allday's. The "mutants" we construct are obtained from compactified representations and involve Hopf bundles in a crucial way.