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

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 .