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

Algebraic & Geometric Topology 22 (2022) 3059–3082
##### Abstract

The cohomology groups of a closed manifold $M$ can be reconstructed using the gradient flow of a Morse–Smale function $f:M\to ℝ$. 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 $M$. Witten showed that these inequalities can be deduced analytically by studying the asymptotic behavior of the deformed Laplacian operator. Adopting Witten’s approach, we provide an analytic proof for the so-called equivariant Morse inequalities when the underlying manifold is acted upon by the Lie group $\mathbb{𝕋}={S}^{1}$, and the Morse function $f$ is invariant with respect to this action.

##### Keywords
equivariant cohomology, Morse inequalities, Witten deformation
##### Mathematical Subject Classification 2010
Primary: 57R18, 57R99
##### Publication
Revised: 22 May 2021
Accepted: 13 September 2021
Published: 30 January 2023
##### Authors
 Mostafa E Zadeh Department of Mathematical Sciences Sharif University of Technology Tehran Iran Reza Moghadasi Department of Mathematical Sciences Sharif University of Technology Tehran Iran