Counting excellent discrete Morse functions on compact orientable surfaces

We obtain the number of non-homologically equivalent excellent discrete Morse functions defined on compact orientable surfaces. This work is a continuation of the study which has been done in [2, 4] for graphs

Perfect discrete Morse functions on 2-complexes

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

Structural aspects of the non-uniformly continuous functions and the unbounded functions within C(X)

We prove in this paper that if a metric space supports a real continuous function which is not uniformly continuous then, under appropriate mild assumptions, there exists in fact a plethora of such functions, in both topological and algebraical senses. Corresponding results are also obtained concerning unbounded continuous functions on a non-compact metrizable space.Plan Andaluz de Investigación (Junta de Andalucía)Ministerio de Economía y Competitividad (MINECO). Españ

Critical elements of proper discrete Morse functions

The aim of this paper is to study the notion of critical element of a proper discrete Morse function defined on non-compact graphs and surfaces. It is an extension to the non-compact case of the concept of critical simplex which takes into account the monotonous behaviour of a function at the ends of a complex. We show how the number of critical elements are related to the topology of the complex.Plan Nacional de Investigación (Ministerio de Educación y Ciencia

Morse–Bott theory on posets and a homological Lusternik–Schnirelmann theorem

We develop Morse–Bott theory on posets, generalizing both discrete Morse–Bott theory for regular complexes and Morse theory on posets. Moreover, we prove a Lusternik– Schnirelmann theorem for general matchings on posets, in particular, for Morse–Bott functions

Elementos de la teoría de grupoides y algebroides

Esta monografía nace con la idea de servir de referencia básica a todas aquellas personas que necesiten la Teoría de Grupoides y Algebroides, bien para continuar en sus investigaciones sobre estos mismos objetos o bien para servirse de ellos en el estudio de otros diferentes

Simplicial Lusternik-Schnirelmann category

The simplicial LS-category of a finite abstract simplicial complex is a new invariant of the strong homotopy type, defined in purely combinatorial terms. We prove that it generalizes to arbitrary simplicial complexes the well known notion of arboricity of a graph, and that it allows to develop many notions and results of algebraic topology which are costumary in the classical theory of Lusternik-Schnirelmann category. Also we compare the simplicial category of a complex with the LS-category of its geometric realization and we discuss the simplicial analogue of the Whitehead formulation of the LS-category

Funciones de Morse discretas sobre complejos infinitos

Esta memoria está dedicada a la extensión para complejos simpliciales infinitos de los conceptos y resultados de la teoría de Morse discreta ya estudiados en el caso finito. En un principio, dicho estudio se centrará en el caso de los 1-complejos infinit

Desigualdades de Morse generalizadas sobre grafos

Junta de Andalucí