Fixed Point Theory for Digital k -Surfaces and Some Remarks on the Euler Characteristics of Digital Closed Surfaces

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, 2019, vol. 7, issue 12, 1-19

Abstract: The present paper studies the fixed point property ( FPP ) for closed k -surfaces. We also intensively study Euler characteristics of a closed k -surface and a connected sum of closed k -surfaces. Furthermore, we explore some relationships between the FPP and Euler characteristics of closed k -surfaces. After explaining how to define the Euler characteristic of a closed k -surface more precisely, we confirm a certain consistency of the Euler characteristic of a closed k -surface and a continuous analog of it. In proceeding with this work, for a simple closed k -surface in Z 3 , say S k , we can see that both the minimal 26-adjacency neighborhood of a point x ∈ S k , denoted by M k ( x ) , and the geometric realization of it in R 3 , denoted by D k ( x ) , play important roles in both digital surface theory and fixed point theory. Moreover, we prove that the simple closed 18-surfaces M S S 18 and M S S 18 ′ do not have the almost fixed point property ( AFPP ). Consequently, we conclude that the triviality or the non-triviality of the Euler characteristics of simple closed k -surfaces have no relationships with the FPP in digital topology. Using this fact, we correct many errors in many papers written by L. Boxer et al.

Keywords: fixed point property; almost (approximate) fixed point property; digital surface; digital connected sum; geometric realization; Euler characteristic; minimal (3 n ? 1)-neighborhood; digital topology (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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