CHAPTER 2
Algebraic Discrete Morse Theory
In this chapter we describe an algebraic version of discrete Morse theory. Dis-
crete Morse theory was developed in the mid 90’s by Robin Forman (see [17],
[18] and [36] for extensions and an alternative approach). Starting from a regular
CW-complex, discrete Morse theory provides a combinatorial tool for construction
a homotopy equivalent CW-complex with fewer cells. Our theory is an algebraic
version of Forman’s theory which operates on chain complexes rather than on CW-
complexes. It is generalization of results from [6] and an almost identical theory has
been developed independently by Sk¨ oldberg [44] and by Kozlov [33]. Our applica-
tions require a slightly more general setting than the one covered in [44] and [33].
Therefore, we give a detailed exposition of the theory here and provide proofs in
Appendix B. We would like to note that meanwhile Emil Sk¨ oldberg has developed
an even more more general version of discrete Morse theory [46] that no longer
requires free chain complexes.
Let R be a ring and C• = (Ci, ∂i)i≥0 be a chain complex of free R-modules Ci.
We choose a basis X =
i≥0
Xi such that Ci
c∈Xi
R c. From now on we write
the differentials ∂i with respect to the basis X in the following form:
∂i :



Ci Ci−1
c ∂i(c) =
c ∈Xi−1
[c : c ] · c .
Given the complex C• and the basis X, we construct a directed, weighted graph
G(C•) = (V, E). The set of vertices V of G(C•) is the basis V = X and the set E
of (weighted) edges is given by the rule
(c, c , [c : c ]) E :⇔ c Xi, c Xi−1, and [c : c ] = 0.
We often omit the weight and write c c to denote an edge in E. Also by abuse
of notation we write e G(C•) to indicate that e is an edge in E.
Definition 2.1. A finite subset M E of the set of edges is called an acyclic
matching or Morse matching if it satisfies the following three conditions:
(1) (Matching) Each vertex v V lies in at most one edge e M.
(2) (Invertibility) For all edges (c, c , [c : c ]) M the weight [c : c ] lies in the
center of R and is a unit in R.
(3) (Acyclicity) The graph GM(V, EM) has no directed cycles, where EM is
given by
EM := (E \ M) c , c,
−1
[c : c ]
with (c, c , [c : c ]) M .
7
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