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

Oliver Goertsches, Leopold Zoller

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.

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

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.

Paper Structure

This paper contains 5 sections, 27 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Orbit type stratification
  4. Cohomology
  5. Remarks on the integral case

Key Result

Theorem 1

The rational equivariant cohomology of a compact $T$-space $X$ encodes the subposet $\overline{\chi}$ of ramified elements in the connected orbit type stratification, together with the function $\overline{\lambda}:\overline{\chi}\to \{\textrm{connected subgroups of }T\}$ that sends an element $C\in\

Theorems & Definitions (67)

  • Theorem
  • Corollary
  • Proposition
  • Corollary
  • Theorem
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Lemma 2.4
  • Definition 2.5
  • ...and 57 more