Mapping Cone and Morse Theory

TL;DR

This paper introduces a Morse theoretical framework for mapping cone cohomology on smooth manifolds, providing a new computational approach and bounds related to critical points and cohomology dimensions.

Contribution

It develops a Morse complex for the mapping cone cohomology, proving its invariance and establishing inequalities relating critical points to cohomological properties.

Findings

Morse complex for mapping cone cohomology is isomorphic to the original cohomology.

The cohomology is independent of the Riemannian metric and Morse function.

Derived bounds on the dimension of cohomology and critical points.

Abstract

On a smooth manifold, we associate to any closed differential form a mapping cone complex. The cohomology of this mapping cone complex can vary with the de Rham cohomology class of the closed form. We present a novel Morse theoretical description for the mapping cone cohomology. Specifically, we introduce a Morse complex for the mapping cone complex which is generated by pairs of critical points with the differential defined by gradient flows and an integration of the closed form over spaces of gradient flow lines. We prove that the cohomology of our cone Morse complex is isomorphic to the mapping cone cohomology and hence independent of both the Riemannian metric and the Morse function used to define the complex. We also obtain sharp inequalities that bound the dimension of the mapping cone cohomology in terms of the number of Morse critical points and the properties of the specified closed form. Our results are widely applicable, especially for any manifold equipped with a geometric structure described by a closed differential form. We also obtain a bound on the difference between the number of Morse critical points and the Betti numbers.