Eladio Domínguez

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.

Digital homotopy with obstacles

In (Ayala et al. (Discrete Appl. Math. 125 (1) (2003) 3) it was introduced the notion of a digital fundamental group π1d(O/S; σ) for a set of pixels O in relation to another set S which plays the role of an obstacle. This notion intends to be a generalization of the digital fundamental groups of both digital objects and their complements in a digital space. However, the suitability of this group was only checked for digital objects in that paper. As a sequel, we extend here the results in Ayala et al. (2003) for complements of objects. More precisely, we prove that for arbitrary digital spaces the group π1d(O/S; σ) maps onto the usual fundamental group of the difference of continuous analogues |AO ∪ S| - |AS|. Moreover, this epimorphism turns to be an isomorphism for a large class of digital spaces including most of the examples in digital topology.

Digital homotopy with obstacles.

Abstract As a sequel of [4], this paper is devoted to the computation of the digital fundamental group πd1(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 πd1(O/S;σ) maps onto the usual fundamental group of the difference of continuous analogues ∣n A O∪S∣ − ∣n A S∣. 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.

EXPONENTIAL LAW FOR UNIFORMLY CONTINUOUS PROPER MAPS

The purpose of this note is to prove the exponential law for uniformly continuous proper maps.