Discrete Morse Theory for Posets

When the set of cells of a CW complex is given by means of a combinatorial enumeration, and the cell attachment maps are not too complicated, for instance if the CW complex in question is regular, it is natural to attempt to use the standard notion of cellular collapse to simplify the considered topological space, while preserving its homotopy t3rpe. 

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Beyond the encoding of all allowed collapsing orders as the set of linear extensions of the universal object U (P, M), viewing the posets with small fibers as the central notion of the combinatorial part of discrete Morse theory is also invaluable for the structural explanation of a standard way to construct acyclic matchings as unions of acyclic matchings on fibers of a poset map. 

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