Greedy Morse matchings and discrete smoothness
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Discrete Morse theory emerged as an essential tool for computational geometry and topology. Its core structures are discrete gradient fields, defined as acyclic matchings on a complex C, from which topological and geometrical informations of C can be efficiently computed, in particular its homology or Morse-Smale decomposition. Given a function f sampled on C, it is possible to derive a discrete gradient that mimics the dynamics of f. Many such constructions are based on some variant of a greedy pairing of adjacent cells, given an appropriate weighting. However, proving that the dynamics of f is correctly captured by this process is usually intricate. This work introduces the notion of discrete smoothness of the pair (f,C), as a minimal sampling condition to ensure that the discrete gradient is geometrically faithful to f. More precisely, a discrete gradient construction from a function f on a polyhedron complex C of any dimension is studied, leading to theoretical guarantees prior to the discrete smoothness assumption. Those results are then extended and completed for the smooth case. As an application, a purely combinatorial proof that all CAT(0) cube complexes are collapsible is given.
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