UTILITY OF DIGITAL COVERING THEORY

Abstract

Abstract. Various properties of digital covering spaces have beensubstantially used in studying digital homotopic properties of dig-ital images. In particular, these are so related to the study of adigital fundamental group, a classi cation of digital images, an au-tomorphism group of a digital covering space and so forth. Thegoal of the present paper, as a survey article, to speak out utilityof digital covering theory. Besides, the present paper recalls thatthe papers [1, 4, 30] took their own approaches into the study ofa digital fundamental group. For instance, they consider the dig-ital fundamental group of the special digital image (X;4), whereX := SC 2;84 which is a simple closed 4-curve with eight elements inZ 2 , as a group which is isomorphic to an in nite cyclic group suchas (Z;+). In spite of this approach, they could not propose any dig-ital topological tools to get the result. Namely, the papers [4, 30]consider a simple closed 4 or 8-curve to be a kind of simple closedcurve from the viewpoint of a Hausdor topological structure, i.e. acontinuous analogue induced by an algebraic topological approach.However, in digital topology we need to develop a digital topologi-cal tool to calculate a digital fundamental group of a given digitalspace. Finally, the paper [9] rstly developed the notion of a digitalcovering space and further, the advanced and simpli ed version wasproposed in [21]. Thus the present paper refers the history and theprocess of calculating a digital fundamental group by using varioustools and some utilities of digital covering spaces. Furthermore, wedeal with some parts of the preprint [11] which were not publishedin a journal (see Theorems 4.3 and 4.4). Finally, the paper suggestsan ecient process of the calculation of digital fundamental groupsof digital images.Received August 18, 2014. Accepted August 28, 2014.2010 Mathematics Subject Classi cation. 55Q70, 52Cxx, 55P15, 68R10, 68U05.Key words and phrases. digital topology, digital product, k-homotopic thinning,normal adjacency, S-compatible adjacency, digital covering space, C-property, S-property.