Teoría de morse discreta

Abstract

Since it was introduced by Marston Morse in the 1920s, Morse Theory has been one of the most powerful and useful tool for the study of smooth manifolds. Basically, it allows to describe the topology of a manifold in terms of the cellular decomposition generated by the critical points of a scalar smooth map de ned on it. In the 1990s Robin Forman developed a combinatorial analog to Morse theory that turned out to be a fruitful and e cient method for studying the homotopy type and homology groups of discrete objects, such as simplicial and cellular complexes. The essential components of Forman's discrete Morse theory are the discrete Morse function and critical símplices. A discrete Morse function assigns values to símplices such that higher-dimensional símplices have higher values than lower-dimensional símplices, with at most one exception, locally, at each simplex. A critical simplex is one near which no such exception occurs. The main reason why these de nitions seem to be quite di erent from those of di erential geometry is that they correspond to a natural and easily representable notion of induced discrete vector eld which includes the idea of ow lines ressembling the corresponding notion for smooth vector elds. Roughly speaking, the discrete vector eld induced by a discrete Morse function describe a collapsing process, and is a pairing of the complex, formed by pairs of símplices of two consecutives dimensions on which the function inverts the order given by face inclusion. Then, those símplices that do not belong to any pair are precisely the critical símplices of the map and a combinatorial gradient path is a connected sequence of the pairs on which the function is decreasing. In this work we present the basic notions and results of discrete Morse theory, with emphasis in two aspects: it can be interpreted under a graph-theoretical point of view as a matching in the Hasse diagram of its domain and the special role played by the optimal discrete Morse functions, that is, those with as less critical símplices as possible. For simplicity, we have omitted the proofs of the more technical results that can be found in detail in the cited references.