CHAPTER 2 Algebraic Discrete Morse Theory In this chapter we describe an algebraic version of discrete Morse theory. Dis- crete Morse theory was developed in the mid 90’s by Robin Forman (see [17], [18] and [36] for extensions and an alternative approach). Starting from a regular CW-complex, discrete Morse theory provides a combinatorial tool for construction a homotopy equivalent CW-complex with fewer cells. Our theory is an algebraic version of Forman’s theory which operates on chain complexes rather than on CW- complexes. It is generalization of results from [6] and an almost identical theory has been developed independently by Sk¨ oldberg [44] and by Kozlov [33]. Our applica- tions require a slightly more general setting than the one covered in [44] and [33]. Therefore, we give a detailed exposition of the theory here and provide proofs in Appendix B. We would like to note that meanwhile Emil Sk¨ oldberg has developed an even more more general version of discrete Morse theory [46] that no longer requires free chain complexes. Let R be a ring and C• = (Ci, ∂i)i≥0 be a chain complex of free R-modules Ci. We choose a basis X = i≥0 Xi such that Ci c∈Xi R c. From now on we write the differentials ∂i with respect to the basis X in the following form: ∂i : Ci Ci−1 c ∂i(c) = c ∈Xi−1 [c : c ] · c . Given the complex C• and the basis X, we construct a directed, weighted graph G(C•) = (V, E). The set of vertices V of G(C•) is the basis V = X and the set E of (weighted) edges is given by the rule (c, c , [c : c ]) E :⇔ c Xi, c Xi−1, and [c : c ] = 0. We often omit the weight and write c c to denote an edge in E. Also by abuse of notation we write e G(C•) to indicate that e is an edge in E. Definition 2.1. A finite subset M E of the set of edges is called an acyclic matching or Morse matching if it satisfies the following three conditions: (1) (Matching) Each vertex v V lies in at most one edge e M. (2) (Invertibility) For all edges (c, c , [c : c ]) M the weight [c : c ] lies in the center of R and is a unit in R. (3) (Acyclicity) The graph GM(V, EM) has no directed cycles, where EM is given by EM := (E \ M) c , c, −1 [c : c ] with (c, c , [c : c ]) M . 7
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