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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