Morse inequalities for noncompact manifolds

TL;DR

This paper extends Morse inequalities to noncompact manifolds with group actions, relating critical points to $L^2$-Betti numbers and group functions, even when Morse functions lack invariance.

Contribution

It introduces Morse inequalities for noncompact manifolds with group actions without requiring Morse functions to be invariant.

Findings

Morse inequalities relate critical points to $L^2$-Betti numbers.

Inequalities incorporate functions on the acting group.

Results apply to manifolds with cocompact, properly discontinuous group actions.

Abstract

We establish Morse inequalities for a noncompact manifold with a cocompact and properly discontinuous action of a discrete group, where Morse functions are not necessarily invariant under the group action. The inequalities are given in terms of the $L^2$-Betti numbers and functions on the acting group which describe rough configurations of critical points of a Morse function.