Freeness of equivariant cohomology and mutants of compactified representations

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.