Equivariant Morse-Bott cohomology through stabilization

TL;DR

This paper develops a Morse-theoretic approach to equivariant cohomology on manifolds with Lie group actions, using stabilization to handle transversality and orientability issues.

Contribution

It introduces a stabilization technique to produce stable Morse-Bott functions, enabling generic invariant metrics to satisfy transversality and orientability in equivariant cohomology.

Findings

Stable invariant Morse-Bott functions can be achieved via small equivariant perturbations.

The method ensures equivariant transversality and orientability assumptions are realizable.

The approach simplifies the construction of equivariant cohomology using Morse theory.

Abstract

For closed manifolds with compact Lie group actions, we study Austin-Braam's Morse-theoretic construction of Borel equivariant cohomology using the technique of stabilization. We show that a $C^1$-small equivariant perturbation produces stable invariant Morse-Bott functions. This allows us to realize the equivariant transversality and orientability assumptions in Austin-Braam's framework by choosing generic invariant Riemannian metrics.