Reconstructing the orbit type stratification of a torus action from its equivariant cohomology

TL;DR

This paper explores how the equivariant cohomology of a space with a torus action reveals the structure of its orbit type stratification, especially focusing on the encoded poset and cohomological data.

Contribution

It demonstrates that equivariant cohomology encodes the stratification's subposet of ramified elements and, under certain conditions, the cohomology of individual strata.

Findings

Equivariant cohomology encodes the subposet of ramified elements.

For equivariantly formal actions, cohomological data of stratification is captured.

In smooth cases, equivariant cohomology of the manifold determines that of the strata.

Abstract

We investigate what information on the orbit type stratification of a torus action on a compact space is contained in its rational equivariant cohomology algebra. Regarding the (labelled) poset structure of the stratification we show that equivariant cohomology encodes the subposet of ramified elements. For equivariantly formal actions, we also examine what cohomological information of the stratification is encoded. In the smooth setting we show that under certain conditions -- which in particular hold for a compact orientable manifold with discrete fixed point set -- the equivariant cohomologies of the strata are encoded in the equivariant cohomology of the manifold.