Morse theory for manifolds with boundary

TL;DR

This paper extends Morse theory to manifolds with boundary, showing how interior critical points can be moved to the boundary and analyzing cobordisms in this context.

Contribution

It introduces a topological method to relocate interior critical points to the boundary and applies this to decompose cobordisms of manifolds with boundary.

Findings

Interior critical points can be moved to the boundary under certain conditions

Cobordisms of manifolds with boundary can be decomposed into product cobordisms

Standard Morse theory results are extended to manifolds with boundary

Abstract

We develop Morse theory for manifolds with boundary. Besides standard and expected facts like the handle cancellation theorem and the Morse lemma for manifolds with boundary, we prove that, under a topological assumption, a critical point in the interior of a Morse function can be moved to the boundary, where it splits into a pair of boundary critical points. As an application, we prove that every cobordism of manifolds with boundary splits as a union of left product cobordisms and right product cobordisms.