Perfect discrete Morse functions on 2-complexes

Abstract

A bst r ac t This paper is focused on the study of perfect discrete Morse functions on a 2-simplicial complex. These are those discrete Morse functions such that the number of critical i-simplices coincides with the i-th Betti number of the complex. In particular, we establish conditions under which a 2-complex admits a perfect discrete Morse function and conversely, we get topological properties of a 2-complex admitting such kind of functions. This approach is more general than the known results in the literature [7], since our study is not restricted to surfaces. These results can be considered as a first step in the study of perfect discrete Morse functions on 3-manifolds.K e y wor ds perfect discrete Morse function, Betti number, simplicial complex. I nt roductionSince it was introduced, Morse theory has been a powerful tool in the study of smooth manifolds by means of differential geometry techniques. Basically, it allows us to describe the topology of a manifold in terms of the cellular decomposition generated by the critical points of a scalar smooth map defined on it.At the end of the last century, R. Forman [4] developed a discrete version of Morse theory that turned out to be a fruitful and efficient method for the study of the topology of discrete objects, such as simplicial and cellular complexes, which play a central role in many different fields of pure and applied mathematics.Essentially, a discrete Morse function on a simplicial complex is a way to assign a real number to each simplex of a complex, without any continuity, in such a way that for each simplex the natural order given by the dimension simplices is respected, except at most in one (co)face of the given simplex. As in the smooth setting, changes in the topology of the level subcomplexes are deeply related to the presence of critical simplices of the function, and the analysis of the evolution of the homology of these complexes can be a very useful tool in computer vision to deal with shape recognition problems by means of topological shape descriptors. In our opinion, there are many advantages of using Forman's theory. First, it can be applied to discrete objects more general than manifolds. Second, it is more suitable in the digital context on topics like pattern recognition, shape classification and recognition, thinning 2D-objects where usually discretized functions are used.Optimal discrete Morse functions has been widely studied in the literature [6,7]. However, this question is not usually considered as an optimization problem in terms of obtaining discrete Morse functions with as less critical simplices as possible. On the contrary, this problem is mainly settled as a problem of perfect discrete Morse functions, that is, those functions satisfyingwhere m i ( f ) is indicating the number of i-critical simplices of f on K and b i ( K ; F ) is the i-th Betti number of M . In this setting is interesting to point out that there are complexes which do not admit perfect discrete Morse functions. It can be explained by two main reasons:...

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“…This is the case of the Bing's house and the Dunce hat complexes, that are contractible but not collapsible (see (Ayala et al, 2010)). …”

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Perfect Discrete Morse Functions on Triangulated 3-Manifolds

“…Notice that the converse is not true. In [2] an example of a 2-complex K with H 1 (K) = Z and not admitting Z-perfect discrete Morse functions is included. In particular, if the first fundamental group of a 2-complex is finite and non-trivial then it does not admit Z-perfect discrete Morse functions.…”

“…This situation strongly contrasts with the smooth case, where the existence of perfect functions on a manifold does not depend on the given triangulation. The main goal of this work, which is the continuation of the paper [2], consists on reducing the problem, initially stated on 3-manifolds, to 2-complexes. It is possible since we prove that a triangulation K of a 3-manifold admits a perfect discrete Morse function if and only if a spine of K admits such kind of function.…”