The Most Refined Axiom for a Digital Covering Space and Its Utilities

Abstract

This paper is devoted to establishing the most refined axiom for a digital covering space which remains open. The crucial step in making our approach is to simplify the notions of several types of earlier versions of local (k0,k1)-isomorphisms and use the most simplified local (k0,k1)-isomorphism. This approach is indeed a key step to make the axioms for a digital covering space very refined. In this paper, the most refined local (k0,k1)-isomorphism is proved to be a (k0,k1)-covering map, which implies that the earlier axioms for a digital covering space are significantly simplified with one axiom. This finding facilitates the calculations of digital fundamental groups of digital images using the unique lifting property and the homotopy lifting theorem. In addition, consider a simple closed k:=k(t,n)-curve with five elements in Zn, denoted by SCkn,5. After introducing the notion of digital topological imbedding, we investigate some properties of SCkn,5, where k:=k(t,n),3≤t≤n. Since SCkn,5 is the minimal and simple closed k-curve with odd elements in Zn which is not k-contractible, we strongly study some properties of it associated with generalized digital wedges from the viewpoint of fixed point theory. Finally, after introducing the notion of generalized digital wedge, we further address some issues which remain open. The present paper only deals with k-connected digital images.

Discrete Group Actions on Digital Objects and Fixed Point Sets by Isok(·)-Actions

“…As mentioned in [39], given 5b). Based on the generalized digital wedge from Definition 8, we obtain the following:…”

“…Based on the concept of the digital topological imbedding above, let us now make the digital wedge (C n,l 1 k ∨ C n,l 2 k , k) more generalized, as follows: Definition 8. [39] Given two digital images (X, k 1 ) in Z n 1 and (Y, k 2 ) in Z n 2 , where k 1 := k(t 1 , n 1 ) and k 2 := k(t 2 , n 2 ), take n := max{n 1 , n 2 } and t := max{t 1 , t 2 }. Then, we define a digital wedge of (X, k 1 ) and (Y, k 2 ) in Z n with a k-adjacency of Z n , where k := k(t, n), denoted by (X ∨ Y, k), as one point union of the certain digital images (X , k 1 := k(t 1 , n)) and (Y , k 2 := k(t 2 , n)) in Z n satisfying the following properties.…”

“…)), we have recently studied the following cases, l i ∈ N 1 , i ∈ {1, 2} [39] and l 1 ∈ N 0 , l 2 ∈ N 1 (or l 1 ∈ N 1 , l 2 ∈ N 0 ) [39]. Thus, for the case of l i ∈ N 0 , i ∈ {1, 2}, the study of F(Con k (C n,l 1 k ∨ C n,l 2 k )) still remains.…”