The Morse-Novikov theory of circle-valued functions and noncommutative localization

TL;DR

This paper develops a noncommutative localization approach to extend Morse theory for circle-valued functions, providing new bounds on critical points and generalizing Novikov inequalities.

Contribution

It introduces a novel noncommutative localization method to construct a generalized Morse complex for circle-valued functions, extending classical results.

Findings

Constructed a chain complex counting critical points of circle-valued Morse functions.

Derived new topological lower bounds on the number of critical points.

Generalized Novikov inequalities to broader settings.

Abstract

We use noncommutative localization to construct a chain complex which counts the critical points of a circle-valued Morse function on a manifold, generalizing the Novikov complex. As a consequence we obtain new topological lower bounds on the minimum number of critical points of a circle-valued Morse function within a homotopy class, generalizing the Novikov inequalities.