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

Author

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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)

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 .

Suggested Citation

  • Sang-Eon Han, 2020. "Digital k -Contractibility of an n -Times Iterated Connected Sum of Simple Closed k -Surfaces and Almost Fixed Point Property," Mathematics, MDPI, vol. 8(3), pages 1-23, March.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:3:p:345-:d:328228
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    References listed on IDEAS

    as
    1. Han, Sang-Eon, 2019. "Estimation of the complexity of a digital image from the viewpoint of fixed point theory," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 236-248.
    2. Jeong Min Kang & Sang-Eon Han & Sik Lee, 2019. "The Fixed Point Property of Non-Retractable Topological Spaces," Mathematics, MDPI, vol. 7(10), pages 1-12, September.
    Full references (including those not matched with items on IDEAS)

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