Dominic Weiller's 2015 honours thesis in mathematics, on "Smooth and Discrete Morse theory", supervised by Dr. Joan Licata
, is available here.
Morse theory is a powerful tool that uses generic functions on a manifold to provide information
about the manifold. Morse theory has been used to prove important results in the study
of smooth manifolds such as the h-cobordism theorem and generalised Poincare conjecture
for dimensions greater than four [22, 26]. Recently, the ideas of Morse theory have been
adapted for use with CW complexes . This provides a tool for analysing these complexes,
and in particular, a secondary way to study smooth manifolds, by first giving them a cell
Chapter I reviews the basics of smooth Morse theory that will be needed for later chapters.
We predominantly follow the approach of Milnor  and Nicolaescu , but with a few
arguments and motivations shown more explicitly. The argument showing the weak Morse
inequalities at the end of the chapter is original, though we suspect a similar argument must
have existed previously.
Chapter II highlights the the utility that being able to find Morse functions with specific
properties can have, through a relatively simple classification of surfaces. We follow the
approach of Champanerkar, Kumar, and Kumaresan  but fix an error in their proof; this
is given by Lemma II.1.3. We also discuss the effect that modifying a Morse function on a
surface has on the structure of sublevel sets; this discussion is inspired by the ideas in 
but addresses issues not covered there.
The latter chapters are predominately focused on discussing discrete Morse theory and
its relationship with smooth Morse theory. Chapter III introduces the the theory discrete
Morse functions developed by Robin Forman  in the 1990s. This theory extends many
ideas from smooth Morse theory to the context of CW complexes. Theorem III.2.4 and the
following corollary provide an original description of a process using a discrete Morse gradient
to collapse a CW complex. This is used implicitly in the literature, though we could not find
a similar explanation of this technique.
Section 4 of Chapter III discusses some similarities and differences in the collections of
Morse or discrete Morse functions that an object supports. In particular we present an
original argument that all discrete Morse functions on a CW complex are homotopic to one
another, through a paths of discrete Morse functions; the analogous statement is not true in
the smooth case.
Particularly interesting is the ability to take any smooth Morse function and induce a
discrete Morse function with similar properties by triangulating the manifold; this is the
content of Chapter IV. We discuss two methods for doing this, one due to Benedetti 
and the other by Gallais . While presenting Benedetti’s approach we discuss shellable
simplicial complexes and include a few original arguments concerning them. In particular
we present original arguments that show the boundary of a simplex is shellable and that a
shellable complex is endocollapsible; these results are mentioned in the literature, but we
could not find proofs as accessable as the ones we provide.
Finally, Chapter V discusses the homology theory of discrete Morse theory introduced by
Forman , highlighting the striking similarity to the homology of smooth Morse theory.
We present an original proof that the boundary map in fact gives us a chain complex; this
extends the proof of Gallais  from the case of simplicial complexes to CW complexes. We
also show that the discrete Morse homology (for CW complexes) is isomorphic to singular
homology; this is a minor extension of the proof given by Gallais for simplicial complexes.