Abstract
This paper relates the recent theory of discrete Morse functions due to Forman (Adv. in Math. 134 (1998) 90–145) and combinatorial decompositions such as shellability, which are known to have many useful applications within combinatorics. First, we present the basic aspects of discrete Morse theory for regular cell complexes in terms of the combinatorial structure of their face posets. We introduce the notion of a generalized shelling of a regular cell complex and describe how to construct a discrete Morse function associated with such a decomposition. An application of Forman's theory gives us generalizations of known results about the homotopy properties of shellable complexes. We also discuss an application to a set of complexes related to matroids.
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