Exact sequences for equivariantly formal spaces

TL;DR

This paper generalizes the Atiyah-Bredon exact sequence for equivariantly formal spaces, extending its applicability to various coefficients and actions without fixed points, thereby broadening the theoretical framework of equivariant cohomology.

Contribution

It introduces a generalized exact sequence for equivariantly formal spaces with broader coefficients and actions without fixed points, extending previous results.

Findings

Generalized Atiyah-Bredon sequence for various coefficients

Extended sequence applicable to fixed point free actions

Broadened theoretical framework for equivariant cohomology

Abstract

Let T be a torus. We present an exact sequence relating the relative equivariant cohomologies of the skeletons of an equivariantly formal T-space. This sequence, which goes back to Atiyah and Bredon, generalizes the so-called Chang-Skjelbred lemma. As coefficients, we allow prime fields and subrings of the rationals, including the integers. We extend to the same coefficients a generalization of this "Atiyah-Bredon sequence" for actions without fixed points which has recently been obtained by Goertsches and Toeben.