Ultra regular covering space and its automorphism group

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Abstract. The present paper investigates some properties of the digital homology in [1,4,5] and points out some unclearness of the definition of a digital homology and further, suggests a suitable form of a digital homology. Finally, we calculate a digital homology group and a relative digital homology group of some digital wedge sums. Finally, the paper corrects some errors in [6]. In the present paper all digital images (X, k) are assumed to be non-empty and k-connected.

Section: Preliminariesmentioning

confidence: 99%

“…(2.1) Concretely, these k(m, n)-adjacency relations of Z n are determined according to the number m ∈ N [7] (see also [13]). …”

Section: Preliminariesmentioning

confidence: 99%

Digital Homology Groups of Digital Wedge Sums

Kang¹,

Han²

2016

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“…To study digital topological properties of a digital image (X, k), we have used various tools such as digital fundamental groups [2,13,16], digital covering spaces [7,8,11,12,14,15,18,19], digital homotopy equivalences [5,24] and digital k-surface structures [12]. A (binary) digital image (X, k) can be regarded as a subset X ⊂ Z n with one of the k-adjacency relations of Z n (or an adjacency graph).…”

Section: Preliminariesmentioning

confidence: 99%

REMARKS ON SIMPLY k-CONNECTIVITY AND k-DEFORMATION RETRACT IN DIGITAL TOPOLOGY

Han¹

2014

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Abstract. To study a deformation of a digital space from the viewpoint of digital homotopy theory, we have often used the notions of a weak k-deformation retract [20] and a strong k-deformation retract [10,12,13]. Thus the papers [10,12,13,16] firstly developed the notion of a strong k-deformation retract which can play an important role in studying a homotopic thinning of a digital space. Besides, the paper [3] deals with a k-deformation retract and its homotopic property related to a digital fundamental group. Thus, as a survey article, comparing among a k-deformation retract in [3], a strong k-deformation retract in [10,12,13], a weak deformation k-retract in [20] and a digital k-homotopy equivalence [5, 24], we observe some relationships among them from the viewpoint of digital homotopy theory. Furthermore, the present paper deals with some parts of the preprint [10] which were not published in a journal (see Proposition 3.1). Finally, the present paper corrects Boxer's paper [3] as follows: even though the paper [3] referred to the notion of a digital homotopy equivalence (or a same k-homotopy type) which is a special kind of a k-deformation retract, we need to point out that the notion was already developed in [5] instead of [3] and further corrects the proof of Theorem 4.5 of Boxer's paper [3] (see the proof of Theorem 4.1 in the present paper). While the paper [4] refers some properties of a deck transformation group (or an automorphism group) of digital covering space without any citation, the study was early done by Han in his paper (see the paper [14]).

“…For some digital images, their digital fundamental groups need not be equal to each other [14]. In relation to the calculation of digital fundamental groups of digital images proposed in the papers [1,4,30], we have used various tools such as a digital isomorphism [3,10,26], a digital homotopy equivalence [6,16,27], a digital covering space [8,9,11,12,13,14,15,16,17,18,20,21,28] and an elementary k-deformation [30,32]. More precisely, the notion of a digital fundamental group was originated by Khalimsky [29].…”

Section: Introductionmentioning

confidence: 99%

“…However, in digital topology we need to develop a digital topological tool to calculate a digital fundamental group of a given digital space. Finally, the paper [9] firstly developed the notion of a digital covering space and further, the advanced and simplified version was proposed in [21]. Thus the present paper refers the history and the process of calculating a digital fundamental group by using various tools and some utilities of digital covering spaces.…”

mentioning

confidence: 99%

Utility of Digital Covering Theory

Han¹,

Lee²

2014

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Abstract. Various properties of digital covering spaces have been substantially used in studying digital homotopic properties of digital images. In particular, these are so related to the study of a digital fundamental group, a classification of digital images, an automorphism group of a digital covering space and so forth. The goal of the present paper, as a survey article, to speak out utility of digital covering theory. Besides, the present paper recalls that the papers [1,4,30] took their own approaches into the study of a digital fundamental group. For instance, they consider the digital fundamental group of the special digital image (X, 4), wherewhich is a simple closed 4-curve with eight elements in Z 2 , as a group which is isomorphic to an infinite cyclic group such as (Z, +). In spite of this approach, they could not propose any digital topological tools to get the result. Namely, the papers [4,30] consider a simple closed 4 or 8-curve to be a kind of simple closed curve from the viewpoint of a Hausdorff topological structure, i.e. a continuous analogue induced by an algebraic topological approach. However, in digital topology we need to develop a digital topological tool to calculate a digital fundamental group of a given digital space. Finally, the paper [9] firstly developed the notion of a digital covering space and further, the advanced and simplified version was proposed in [21]. Thus the present paper refers the history and the process of calculating a digital fundamental group by using various tools and some utilities of digital covering spaces. Furthermore, we deal with some parts of the preprint [11] which were not published in a journal (see Theorems 4.3 and 4.4). Finally, the paper suggests an efficient process of the calculation of digital fundamental groups of digital images.