A User's Guide to Discrete Morse Theory

Robin Forman

Séminaire lotharingien de combinatoire, Tome 48 (2002-2003)

Résumé

A number of questions from a variety of areas of mathematics lead one to the problem of analyzing the topology of a simplicial complex. However, there are few general techniques available to aid us in this study. On the other hand, some very general theories have been developed for the study of smooth manifolds. One of the most powerful, and useful, of these theories is Morse Theory.

In this paper we present a combinatorial adaptation of Morse Theory, which we call discrete Morse Theory, that may be applied to any simplicial complex (or more general cell complex).

The goal of this paper is to present an overview of the subject of discrete Morse Theory that is sufficient both to understand the major applications of the theory to combinatorics, and to apply the the theory to new problems. We will not be presenting theorems in their most recent or most general form, and simple examples will often take the place of proofs. We hope to convey the fact that the theory is really very simple, and there is not much that one needs to know before one can become a "user".