José Antonio Vilches

Perfect discrete Morse functions on 2-complexes

Highlights? We get conditions under which a 2-complex admits a perfect discrete Morse function. ? We study the topology of comlpexes admitting such kind of functions. ? These results are a first step in the study of these functions on 3-manifolds. 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 ith 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 necessary for a 2-complex admitting such kind of functions. This approach is more general than the known results in the literature (Lewiner et al., 2003), 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.

MORSE INEQUALITIES ON CERTAIN INFINITE 2-COMPLEXES

Using the notion of discrete Morse function introduced by R. Forman for finite cw -complexes, we generalize it to the infinite 2-dimensional case in order to get the corresponding version of the well-known discrete Morse inequalities on a non-compact triangulated 2-manifold without boundary and with finite homology. We also extend them for the more general case of a non-compact triangulated 2-pseudo-manifold with a finite number of critical simplices and finite homology.

Perfect discrete morse functions on triangulated 3-manifolds

This work is focused on characterizing the existence of a perfect discrete Morse function on a triangulated 3-manifold M, that is, a discrete Morse function satisfying that the numbers of critical simplices coincide with the corresponding Betti numbers. We reduce this problem to the existence of such kind of function on a spine L of M, that is, a 2-subcomplex L such that M−Δ collapses to L, where Δ is a tetrahedron of M. Also, considering the decomposition of every 3-manifold into prime factors, we prove that if every prime factor of M admits a perfect discrete Morse function, then M admits such kind of function.

A graph-theoretical approach to cancelling critical elements

Abstract This work is focused on the links between Formanʼs discrete Morse theory and graph theory. More precisely, we are interested on putting the optimization of a discrete Morse function in terms of matching theory. It can be done by describing the process of cancellation of pairs of critical simplices by means of obtaining Morse matchings on the corresponding Hasse diagram with a greater number of edges using the combinatorial notion of transference.