  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.
