A Morse complex on manifolds with boundary

TL;DR

This paper introduces a simplified method for constructing Morse complexes on compact manifolds with boundary, using a Morse-Smale vector field, to compute homology with integer coefficients.

Contribution

It presents a new approach that streamlines existing methods for Morse complexes on manifolds with boundary, applicable in more general geometric contexts.

Findings

Morse complex homology matches the manifold's absolute or relative homology.

The method simplifies previous approaches in specific geometric settings.

Applicable to compact smooth manifolds with boundary using Morse functions and pseudo-gradient vector fields.

Abstract

Given a compact smooth manifold $M$ with non-empty boundary and a Morse function, a pseudo-gradient Morse-Smale vector field adapted to the boundary allows one to build a Morse complex whose homology is isomorphic to the (absolute or relative to the boundary) homology of $M$ with integer coefficients. Our approach simplifies other methods which have been discussed in more specific geometric settings.