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

Mostafa E Zadeh and Reza Moghadasi

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 . 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 𝕋 = S1, 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
References
Publication
Received: 8 May 2014
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