Discrete Homotopy of a Closed k-Surface

Ultra regular covering space and its automorphism group

“…For an adjacency relation k of Z n , a simple k-path with l + 1 elements in Z n is assumed to be an injective sequence (x i ) i∈ [0,l] Z ⊂ Z n such that x i and x j are k-adjacent if and only if either j = i + 1 or i = j + 1 (Kong and Rosenfeld, 1996). If x 0 = x and x l = y, then we say that the length of the simple k-path, denoted by l k (x, y), is the number l. A simple closed k-curve with l elements in Z n , denoted by SC n,l k (Han, 2006b), is the simple k-path (x i ) i∈[0,l−1] Z , where x i and x j are k-adjacent if and only if j = i + 1( mod l) or i = j + 1( mod l) (Kong and Rosenfeld, 1996). In the study of digital continuity and various properties of a digital space (Han, 2006a;2006d), we have often used the following digital k-neighborhood of a point x ∈ X with radius ε ∈ N (Han, 2003) (see also Han, 2005c): For a digital space (X, k) in Z n , the digital k-neighborhood of x 0 ∈ X with radius ε is defined in X to be the following subset of X:…”

“…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).…”

“…Motivated by a regular covering space in algebraic topology (Spanier, 1966), its digital version was established in digital covering theory (Han, 2006b) (see also Han, 2007b), which plays an important role in classifying digital covering spaces (Boxer and Karaca, 2008;Han, 2010a). In algebraic topology, for a circle S 1 the existence problem of its regular covering space has substantially contributed to the study of a covering space in topology (Massey, 1977;Spanier, 1966).…”

“…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.…”