• 호남수학학술지
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• 제30권2호
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• pp.207-217
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• 2008

Digital geometry has strongly contributed to the study of a discrete topological space $X{\subset}{\mathbf{Z}}^n$ with k-adjacency of ${\mathbf{Z}}^n$. As a survey-type article, we review various utilities of digital geometry.

• 대한수학회논문집
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• 제12권3호
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• pp.701-707
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• 1997

In [8], we introduce the relative root Nielsen number N(f;X, A, c) for maps of pairs of spaces $f : (X, A) \to (Y, B)$. From it, we obtain some immediate consequences of the definition and illustrate it by some examples. We consider the question whether there exists a map $g : (X, A) \to (Y, B)$ homotopic to a given map $f : (X, A) \to (Y, B)$ which has precisely N(f;X, A, c) roots, that is, the minimum theorem for N(f;X, A, c).

• 호남수학학술지
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• 제24권1호
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• pp.93-101
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• 2002

The $S^1$-Eule characteristics of X is defined by $\bar{\chi}_{S^1}(X)\;{\in}\;HH_1(ZG)$, where G is the fundamental group of connected finite $S^1$-compact manifold or connected finite $S^1$-finite complex X and $HH_1$ is the first Hochsch ild homology group functor. The purpose of this paper is to find several cases which the $S^1$-Euler characteristic has a homotopic invariant.

• 호남수학학술지
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• 제36권3호
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• pp.519-530
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• 2014

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]).

• 호남수학학술지
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• 제36권3호
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• pp.695-706
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• 2014

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), where X := $SC^{2,8}_4$ which 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.

• 호남수학학술지
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• 제33권4호
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• pp.617-628
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• 2011

In relation to the classification of finite topological spaces the paper [17] studied various properties of finite topological spaces. Indeed, the study of future internet system can be very related to that of locally finite topological spaces with some order structures such as preorder, partial order, pretopology, Alexandroff topological structure and so forth. The paper generalizes the results from [17] so that the paper can enlarge topological and homotopic properties suggested in the category of finite topological spaces into those in the category of locally finite topological spaces including ALF spaces.

• 호남수학학술지
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• 제30권4호
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• pp.589-602
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• 2008

As a survey-type article, the paper reviews various digital topological utilities from digital covering theory. Digital covering theory has strongly contributed to the calculation of the digital k-fundamental group of both a digital space(a set with k-adjacency or digital k-graph) and a digital product. Furthermore, it has been used in classifying digital spaces, establishing almost Van Kampen theory which is the digital version of van Kampen theorem in algebrate topology, developing the generalized universal covering property, and so forth. Finally, we remark on the digital k-surface structure of a Cartesian product of two simple closed $k_i$-curves in ${\mathbf{Z}}^n$, $i{\in}{1,2}$.

• Korean Journal of Mathematics
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• 제7권2호
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• pp.151-158
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• 1999

Let (X, G) be a transformation group and ${\sigma}(X,x_0,G)$ the fundamental group of (X, G). In this paper, we prove that the Reidemeister number $R(f_G)$ for an endomorphism $f_G:(X,G){\rightarrow}(X,G)$ is a homotopy invariant. In particular, when any self-map $f:X{\rightarrow}X$ is homotopic to the identity map, we give some calculation of the lower bound of $R(f_G)$. Finally, we discuss commutativity and product formula for the Reidemeister number $R(f_G)$.

• 대한수학회논문집
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• 제33권1호
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• pp.325-335
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• 2018

We prove every oriented compact cyclic 3-orbifold has a contact structure. There is another proof in the web by Daniel Herr in his uploaded thesis which depends on open book decompositions, ours is independent of that. We define overtwisted contact structures, tight contact structures and Lutz twist on oriented compact cyclic 3-orbifolds. We show that every contact structure in an oriented compact cyclic 3-orbifold contactified by our method is homotopic to an overtwisted structure with the overtwisted disc intersecting the singular locus of the orbifolds. In course of proving the above results we prove a classification result for compact oriented cyclic-3 orbifolds which has not been seen by us in literature before.

• Kyungpook Mathematical Journal
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• 제59권3호
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• pp.563-589
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• 2019

Let W and V be manifolds of dimension m + 2, M a locally flat submanifold of V whose dimension is m. Let $f:W{\rightarrow}V$ be a homotopy equivalence. The problem we study in this paper is the following: When is f homotopic to another homotopy equivalence $g:W{\rightarrow}V$ such that g is transverse regular along M and such that $g{\mid}g^{-1}(M):g^{-1}(M){\rightarrow}M$ is a simple homotopy equivalence? $L{\acute{o}}pez$ de Medrano (1970) called this problem the weak h-regularity problem. We solve this problem applying the codimension two surgery theory developed by the author (1973). We will work in higher dimensions, assuming that $$m{\geq_-}5$$.