Analytic approach to $S^1$-equivariant Morse inequalities

TL;DR

This paper extends Witten's analytic approach to prove $S^1$-equivariant Morse inequalities, linking the topology of manifolds with group actions to the asymptotic analysis of deformed Laplacians.

Contribution

It provides the first analytic proof of $S^1$-equivariant Morse inequalities using Witten's method.

Findings

Established an analytic proof for $S^1$-equivariant Morse inequalities.

Connected the asymptotic behavior of deformed Laplacians with equivariant topology.

Extended Witten's approach to manifolds with circle group actions.

Abstract

It is well known that the cohomology groups of a closed manifold $M$ can be reconstructed using the gradient dynamical of a Morse-Smale function $f\colon M\to \R$. A direct result of this construction are Morse inequalities that provide lower bounds for the number of critical points of $f$ in term of Betti numbers of $M$. These inequalities can be deduced through a purely analytic method by studying the asymptotic behaviour of the deformed Laplacian operator. This method was introduced by E. Witten and has inspired a numbers of great achievements in Geometry and Topology in few past decades. In this paper, adopting the Witten approach, we provide an analytic proof for; the so called; equivariant Morse inequalities when the underlying manifold is acted on by the Lie group $G=S^1$ and the Morse function $f$ is invariant with respect to this action.