The relative cup-length in local Morse cohomology

TL;DR

This paper introduces a new relative cup-length in local Morse cohomology, providing a stronger lower bound on the number of critical points of functions on manifolds than previous absolute bounds.

Contribution

It defines a relative cup-length in local Morse cohomology as a module over the neighborhood's cohomology, enhancing critical point lower bounds.

Findings

The relative cup-length provides a sharper lower bound than the absolute cup-length.

Local Morse cohomology forms a module over the cohomology of the isolating neighborhood.

An example demonstrates the effectiveness of the new lower bound.

Abstract

Local Morse cohomology associates cohomology groups to isolating neighborhoods of gradient flows of Morse functions on (generally non-compact) Riemannian manifolds $M$. We show that local Morse cohomology is a module over the cohomology of the isolating neighborhood, which allows us to define a cup-length relative to the cohomology of the isolating neighborhood that gives a lower bound on the number of critical points of functions on $M$ that are not necessarily Morse. Finally, we illustrate by an example that this lower bound can indeed be stronger than the lower bound given by the absolute cup-length.