On integral Chang-Skjelbred computations with disconnected isotropy groups

TL;DR

This paper extends the Chang-Skjelbred method to compute integral cohomology in spaces with disconnected isotropy groups, broadening its applicability and modifying existing formulas for GKM graph cohomology.

Contribution

It demonstrates that integral cohomology can be derived from the one-skeleton even with disconnected isotropy groups under certain conditions, and introduces a modified GKM formula.

Findings

Integral cohomology is encoded in the one-skeleton with disconnected isotropy groups.

Modified GKM formula for graph cohomology.

Applications to Hamiltonian actions.

Abstract

The Chang-Skjelbred method computes the cohomology of a suitable space with a torus action from its equivariant one-skeleton. We show that, under certain restrictions on the cohomological torsion, the integral cohomology is encoded in the one-skeleton even in the presence of arbitrary disconnected isotropy groups. We provide applications to Hamiltonian actions as well as to the GKM case. In the latter, our results lead to a modification of the GKM formula for graph cohomology.