Digital homotopy with obstacles

Abstract

In (Ayala et al. (Discrete Appl. Math. 125 (1) (2003) 3) it was introduced the notion of a digital fundamental group π 1 d ( 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 π 1 d ( O/ S; σ) maps onto the usual fundamental group of the difference of continuous analogues | A O ∪ S |−| A S| . Moreover, this epimorphism turns to be an isomorphism for a large class of digital spaces including most of the examples in digital topology.