Digital homotopy with obstacles

Abstract

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.