Morse theory for manifolds with boundary

Abstract

We develop Morse theory for manifolds with boundary. Beside standard and expected facts like the handle cancellation theorem and the Morse lemma for manifolds with boundary, we prove that under suitable connectedness assumptions 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 connected manifolds with boundary splits as a union of left product cobordisms and right product cobordisms.