Regular Covering Space in Digital Covering Theory and Its Applications

Abstract

Abstract. As a survey-type article, the paper reviews some results on a regular covering space in digital covering theory. The recent paper [10](see also [12]) established the notion of regular covering space in digital covering theory and studied its various properties. Besides, the papers [14, 16] developed a discrete Deck's transformation group of a digital covering. In this paper we study further their properties. By using these properties, we can classify digital covering spaces. Finally, the paper proposes an open problem.

Ultra regular covering space and its automorphism group

“…As discussed by Han (2007b) (see also Han, 2008a), by using Massey's program (Massey, 1977), we obtain the following: Han, 2007b) (see also Han, 2009b) because the group π k1 (B, b 0 ) operates transitively on the set p −1 (b 0 ) (Han, 2007b) (see also Han, 2008a;2010c). Precisely, consider…”

“…(Han, 2009b) Let us recall that, by Theorem 5(2), the group p −1 (b 0 ) found in the works of Han (2007b;2008a;2008d), related to the assertion of Theorem 1(2), is clearly isomorphic to the factor group π k1 (B, b 0 )/p * π k0 (E, e 0 ) as a right π k1 (B, b 0 )-space, and the papers by Han (2007b;2008a;2008d) , k) has the L S -or L C -property found in the research by Han (2009a;2010b;2006b;2007a;2008d). Of course, by Theorem 1, in this case we clearly observe that the given digital covering map p should be regular so that…”

“…Useful tools from algebraic topology and geometric topology for studying digital topological properties of a (binary) digital space include a digital covering space, a (digital) k-fundamental group, a digital k-surface and so forth. These have been studied in numerous papers (Boxer, 1999;Boxer and Karaca, 2008;Han, 2005b;2005c;2005d;2006a;2006b;2006c;2006d;2007a;2007b;2008a;2008b;2008c;2008d;2009a;2009b;2009c;2010a;2010b;2010c;Malgouyres and Lenoir, 2000;Khalimsky, 1987;Rosenfeld and Klette, 2003).…”

“…Unlike these properties, their digital versions have some intrinsic features (Han, 2006b;2007b;2009a;2009b;2009c;2010a;2010b;2010c;In-Soo Kim and Han, 2008). Boxer and Karaca (2008) as well as Han (2006b;2007b;2008a;2008b;2009b) studied an automorphism group of a radius 2 (digital) covering (E, p, B). In addition, Boxer and Karaca (2008) studied a classification of digital spaces by using the conjugacy class corresponding to a digital covering.…”

“…By using these tools, we can calculate digital fundamental groups of some digital spaces and classify digital covering spaces satisfying a radius 2 local isomorphism (Boxer and Karaca, 2008;Han, 2006b;2008b;2008d;2009b). However, for a digital covering which does not satisfy a radius 2 local isomorphism, the study of a digital fundamental group of a digital space and its automorphism group remains open.…”