Digital k -Contractibility of an n -Times Iterated Connected Sum of Simple Closed k -Surfaces and Almost Fixed Point Property

Sang-Eon Han
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Sang-Eon Han: Department of Mathematics Education, Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonju-City, Jeonbuk 54896, Korea

Mathematics, 2020, vol. 8, issue 3, 1-23

Abstract: The paper firstly establishes the so-called n-times iterated connected sum of a simple closed k -surface in Z 3 , denoted by C k n , k ∈ { 6 , 18 , 26 } . Secondly, for a simple closed 18-surface M S S 18 , we prove that there are only two types of connected sums of it up to 18-isomorphism. Besides, given a simple closed 6-surface M S S 6 , we prove that only one type of M S S 6 ? M S S 6 exists up to 6-isomorphism, where ? means the digital connected sum operator. Thirdly, we prove the digital k -contractibility of C k n : = M S S k ? ? ? M S S k ? n - times , k ∈ { 18 , 26 } , which leads to the simply k -connectedness of C k n , k ∈ { 18 , 26 } , n ∈ N . Fourthly, we prove that C 6 2 and C k n do not have the almost fixed point property ( AFPP , for short), k ∈ { 18 , 26 } . Finally, assume a closed k -surface S k ( ⊂ Z 3 ) which is ( k , k ¯ ) -isomorphic to ( X , k ) in the picture ( Z 3 , k , k ¯ , X ) and the set X is symmetric according to each of x y -, y z -, and x z -planes of R 3 . Then we prove that S k does not have the AFPP . In this paper given a digital image ( X , k ) is assumed to be k -connected and its cardinality | X | ≥ 2 .

Keywords: digital image; digital topology; ( k , k¯ )-isomorphism; FPP; AFPP; digital k -contractibility; digital surface; digital connected sum; simple closed k -surface; (almost) fixed point property; iterated connected sum (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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