CHAPTER 1

Introduction

Discrete Morse theory as developed by Forman [17],[18] allows to construct,

starting from a (regular) CW-complex, a new homotopy equivalent CW-complex

with fewer cells. In this paper we describe and apply an algebraic version of this

theory, which we call algebraic discrete Morse Theory. Analogous theories were

developed by Sk¨ oldberg [44] and Kozlov [33]. We consider chain complexes C• =

(Ci, ∂i)i≥0 of free modules Ci over a ring R. A priori, we always fix a basis of

the Ci – the basis elements play the role of the cells in the topological situation.

Then, by an application of algebraic discrete Morse theory one constructs a new

chain complex of free R-modules such that the homology of the two complexes

coincides. Before we come to the applications of algebraic discrete Morse theory

which will contain the main results of this paper we first spend some time to outline

the philosophy behind the theory.

Our construction looked upon from the point of algebra does the following. We

are searching for collections

{D•i) (

| i ∈ I} of short exact subcomplexes

D•i) (

: 0 → R

∂

→ R → 0

in a given complex C•. We require:

(A1) The direct sum D• =

i∈I

D(i)

is a subcomplex of C•.

Then we calculate the quotient complex

C•

D•

.

We require:

(A2) The complex

C•

D•

is a complex of free R-modules.

Thus, passing from C• to

C•

D•

we cancel out one copy of R in two consecutive

homological degrees for each

D•i), (

i ∈ I. The description of the differential of the

quotient complex is more subtle and will be provided later. Since each

D•i) (

is exact

it follows that D• is exact. Therefore, the homology of the quotient complex equals

the homology of the complex C•.

Now we describe the combinatorial procedure that will reflect the algebraic

construction described above. We proceed as follows: Consider the complex C•

as a directed weighted graph, where the vertex-set is given by the chosen basis of

C•. Two vertices c, c of homological degree i and i − 1 respectively are joined by

a directed edge e : c → c if c appears with non-zero coeﬃcient [c : c ] ∈ R in

the differential of c. The of the edge e : c → c is then given by the coeﬃcient

[c : c ]. Now we choose edges with invertible weights such that the collection of

edges satisfies the following two conditions:

(C1) The edges have pairwise disjoint vertex sets.

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