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

• Main Author: Sang-Eon Han
• Format: Article
• Language: English
• Published: MDPI AG 2020-10-01
• Series: Mathematics
• Subjects: local (k0,k1)-isomorphism; unique lifting property; homotopy lifting theorem; digital covering; digital topological imbedding; generalized digital wedge
• Online Access: https://www.mdpi.com/2227-7390/8/11/1868

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 <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><msub><mi>k</mi><mn>0</mn></msub><mo>,</mo><msub><mi>k</mi><mn>1</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula>-isomorphisms and use the most simplified local <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><msub><mi>k</mi><mn>0</mn></msub><mo>,</mo><msub><mi>k</mi><mn>1</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula>-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 <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><msub><mi>k</mi><mn>0</mn></msub><mo>,</mo><msub><mi>k</mi><mn>1</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula>-isomorphism is proved to be a <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><msub><mi>k</mi><mn>0</mn></msub><mo>,</mo><msub><mi>k</mi><mn>1</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula>-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 <inline-formula><math display="inline"><semantics><mrow><mi>k</mi><mo>:</mo><mo>=</mo><mi>k</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula>-curve with five elements in <inline-formula><math display="inline"><semantics><msup><mrow><mi mathvariant="double-struck">Z</mi></mrow><mi>n</mi></msup></semantics></math></inline-formula>, denoted by <inline-formula><math display="inline"><semantics><mrow><mi>S</mi><msubsup><mi>C</mi><mi>k</mi><mrow><mi>n</mi><mo>,</mo><mn>5</mn></mrow></msubsup></mrow></semantics></math></inline-formula>. After introducing the notion of digital topological imbedding, we investigate some properties of <inline-formula><math display="inline"><semantics><mrow><mi>S</mi><msubsup><mi>C</mi><mi>k</mi><mrow><mi>n</mi><mo>,</mo><mn>5</mn></mrow></msubsup></mrow></semantics></math></inline-formula>, where <inline-formula><math display="inline"><semantics><mrow><mi>k</mi><mo>:</mo><mo>=</mo><mi>k</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>,</mo><mn>3</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mi>n</mi></mrow></semantics></math></inline-formula>. Since <inline-formula><math display="inline"><semantics><mrow><mi>S</mi><msubsup><mi>C</mi><mi>k</mi><mrow><mi>n</mi><mo>,</mo><mn>5</mn></mrow></msubsup></mrow></semantics></math></inline-formula> is the minimal and simple closed <i>k</i>-curve with odd elements in <inline-formula><math display="inline"><semantics><msup><mrow><mi mathvariant="double-struck">Z</mi></mrow><mi>n</mi></msup></semantics></math></inline-formula> which is not <i>k</i>-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 <i>k</i>-connected digital images.