Analytic approach to S1–equivariant Morse inequalities

Zadeh, Mostafa E; Moghadasi, Reza

doi:10.2140/agt.2022.22.3059

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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

DOI: 10.2140/agt.2022.22.3059

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 $𝕋 = S^{1}$ , 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

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

Volume 22, issue 7 (2022)