Discrete Morse Theory

Abstract

This chapter contains a presentation of discrete Morse theory as developed by Robin Forman (see e.g. [20], [21]). This theory allows to combinatorially construct from a given (regular, finite) CW-complex a second CW-complex that is homotopy equivalent to the first but has fewer cells. As the upshot of this chapter we then show that one can use this theory in order to construct minimal free resolutions (see also [3]). Discrete Morse theory has found many more applications in Geometric Combinatorics and other fields of mathematics, we will not be able to speak about them. We refer the reader for example to [29] where most applications of discrete Morse theory to complexes of graphs are reviewed. There are even promising attempts to find real world applications of discrete Morse theory (see [31]) to image analysis.