The cohomology groups of a closed manifold can be reconstructed using the gradient flow of a Morse–Smale function . A direct result of this construction are Morse inequalities that provide lower bounds for the number of critical points of in term of Betti numbers of . 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 , and the Morse function is invariant with respect to this action.
"Analytic approach to –equivariant Morse inequalities." Algebr. Geom. Topol. 22 (7) 3059 - 3082, 2022. https://doi.org/10.2140/agt.2022.22.3059