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