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.
Highlights
In Z3, the concept of closed k-surface was established in References [1,2] and its digital topological characterizations were studied [3,4,5,6,7]
Owing to properties (1)–(4), we prove that MSS18 complete the proof
Since the present paper focuses on the study of several types of connected sums of the simple
Summary
In Z3 , the concept of closed k-surface was established in References [1,2] and its digital topological characterizations were studied [3,4,5,6,7]. Since a digital image ( X, k) can be recognized as a digital k-graph [5,23], we mainly use the digital k-graphical method to study Euler characteristics of a closed k-surface in this paper. Let us recall two types of simple closed 18-surfaces which are pointed 18-contractible, e.g.,. 18-surface (see Figure 1b), i.e., we obtain 0 in this paper, we prove it more precisely, In order to use the pointed 18-contractibility of MSS18 as follows:. Let us further introduce two simple closed k-surface, k ∈ {18, 26}, as follows: MSS18 ≈18 ( MSC8 × {1}) ∪ ( Int( MSC8 ) × {0, 2}) [3,4]. We can conclude that ( T, 6) is pointed 6-contractible relative to any singleton {t} ⊂ T
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