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".

Received: September 15, 2002. Accepted: September 17, 2002. Final version: October 9, 2002.

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