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.
Highlights
Since 2004 [1,2,3,4], the notion of digital covering space has been strongly used in digital topology and digital geometry
The unique path lifting property [3] and the homotopy lifting theorem [2] among others [4,5,6] are key properties which can be used in calculating digital k-fundamental groups of digital images ( X, k), classifying digital images using algebraic invariants [1,2,5], and their applications such as the winding number in a digital topological setting
It turns out that the simplified version of a local (k0, k1 )-isomorphism gives us very strong advantages of making the earlier version of axioms for a digital covering space most simplified. This approach is a key step in addressing the hot issue which was an unsolved problem during the last 20 years. It appears that the most refined local (k0, k1 )-isomorphism of Definition 6 implies the earlier version of axioms for a digital covering space
Summary
Since 2004 [1,2,3,4], the notion of digital covering space has been strongly used in digital topology and digital geometry. Regarding the issue referred to as (Q3), a recent paper [9] tried to establish certain relationships between a (strong) local (k0 , k1 )-isomorphism [2] and a digital (k0 , k1 )-map by using the notions of “conciliator” [9] and the upl-property for a (k0 , k1 )-surjection [2,3] This approach invokes certain redundant results because it took a detour for the work instead of a direct proof using the given hypothesis. Using the notion of digital topological imbedding, we firstly introduce the notion n ,l1 of generalized digital wedge and further, study an alignment of fixed point sets of SCk 1.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
