Defining discrete Morse functions on infinite surfaces

We present an algorithm which defines a discrete Morse function in Forman’s sense on an infinite surface including a study of the minimality of this function.Plan Andaluz de Investigación (Junta de Andalucía

Exponential law for uniformly continous proper maps

The purpose of this note is to prove the exponential law for uniformly continuous proper maps.Comisión Asesora de Investigación Científica y Técnic

The equivariant category of proper G-Spaces

Direccion General de Investigacion Cientifica y Tecnic

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

Lusternik-Schnirelmann invariants in proper homotopy theory

We introduce and study proper homotopy invariants of the Lusternik-Schnirelmann type, p-cat (-), p-Cat(-), and cat e(-) in the category of Γ2-locally compact spaces and proper maps. As an application, Rn (n Φ 3) is characterized as (i) the unique open manifold X with p-Cat(ΛΓ) = 2, or (ii) the unique open manifold with one strong end and p-cat( c) = 2

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

Homotopy in digital spaces

The main contribution of this paper is a new “extrinsic” digital fundamental group that can be readily generalized to define higher homotopy groups for arbitrary digital spaces. We show that the digital fundamental group of a digital object is naturally isomorphic to the fundamental group of its continuous analogue. In addition, we state a digital version of the Seifert-Van Kampen theorem.Dirección General de Investigación Científica y TécnicaDirección General de Enseñanza Superio

A digital index theorem

Proc. of the 7th Int. Workshop on Combinatorial Image Analysis. IWCIA00. Caen. France. July 2000.This paper is devoted to prove a Digital Index Theorem for digital (n − 1)-manifolds in a digital space (Rn, f), where f belongs to a large family of lighting functions on the standard cubical decomposition Rn of the n-dimensional Euclidean space. As an immediate consequence we obtain the corresponding theorems for all (α, β)-surfaces of Kong-Roscoe, with α, β ∈ {6, 18, 26} and (α, β) 6≠(6, 6),(18, 26),(26, 26), as well as for the strong 26-surfaces of Bertrand-Malgouyres.Dirección General de Investigación Científica y TécnicaDirección General de Enseñanza Superio

Digital homotopy with obstacles

As a sequel of [4] Ayala, R., E. Dom´ıguez, A. R. Franc´es and A. Quintero, Homotopy in Digital Spaces, Discrete and Applied Mathematics, To Appear, this paper is devoted to the computation of the digital fundamental group π d 1 (O/S; σ) defined by loops in the digital object O for which the digital object S acts as an “obstacle”. We prove that for arbitrary digital spaces the group π d 1 (O/S; σ) maps onto the usual fundamental group of the difference of continuous analogues |AO∪S | − |AS |. Moreover, we show that this epimorphism turns to be an isomorphism for a large class of digital spaces including most of the examples in digital topology.Dirección General de Enseñanza Superio