8 2. ALGEBRAIC DISCRETE MORSE THEORY For an acyclic matching M on the graph G(C•) = (V, E) we introduce the following notation, which is an adaptation of the notation introduced in [17] to our situation. (1) We call a vertex c ∈ V critical with respect to M if c does not lie in any edge e ∈ M we write Xi M := {c ∈ Xi | c critical } for the set of all critical vertices of homological degree i. (2) We write c ≤ c if c ∈ Xi, c ∈ Xi−1, and [c : c ] = 0. (3) Path(c, c ) is the set of paths from c to c in the graph GM(C•). (4) The weight w(p) of a path p = c1 → ··· → cr ∈ Path(c1, cr) is given by w(c1 → . . . → cr) := r−1 i=1 w(ci → ci+1), w(c → c ) := ⎧ ⎪ ⎪ − 1 [c : c ] , c ≤ c , [c : c ] , c ≤ c. (5) We write Γ(c, c ) = p∈Path(c,c ) w(p) for the sum of weights of all paths from c to c . Now we are in position to define a new complex C• M , which we call the Morse complex of C• with respect to M. The complex C• M = (Ci M , ∂i M )i≥0 is defined by Ci M := c∈XM i R c, ∂i M : ⎧ ⎨ ⎩ CM i → CM i−1 c → c ∈XM i−1 Γ(c, c )c , . Theorem 2.2. C• M is a complex of free R-modules and is homotopy equivalent to the complex C• in particular, for all i ≥ 0 Hi(C•) ∼ Hi(C• M ). The maps defined below are chain homotopies between C• and C• M : f : ⎧ ⎨ ⎩ C• → C•M c ∈ Xi → f(c) := c ∈XM i Γ(c, c )c , g : ⎧ ⎨ ⎩ CM • → C• c ∈ XM i → gi(c) := c ∈Xi Γ(c, c )c . The proof of Theorem 2.2 is given in Appendix B. Note that if C• is the cellular chain complex of a regular CW-complex and X is the set of cells of the regular CW-complex, then algebraic discrete Morse theory is the part of Forman’s [17] discrete Morse theory which describes the impact of a discrete Morse matching on the cellular chain complex of the CW-complex.

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