Analytic approach to S1–equivariant Morse inequalities

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.