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

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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 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.

Suggested Citation

  • Sang-Eon Han, 2019. "Fixed Point Theory for Digital k -Surfaces and Some Remarks on the Euler Characteristics of Digital Closed Surfaces," Mathematics, MDPI, vol. 7(12), pages 1-19, December.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:12:p:1244-:d:298514
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    References listed on IDEAS

    as
    1. Sang-Eon Han, 2019. "Remarks on the Preservation of the Almost Fixed Point Property Involving Several Types of Digitizations," Mathematics, MDPI, vol. 7(10), pages 1-14, October.
    2. 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.
    Full references (including those not matched with items on IDEAS)

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