• Honam Mathematical Journal
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• v.37 no.4
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• pp.595-608
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• 2015

In this paper, as a survey paper, we review many works related to fixed point theory for digital spaces using Lefschetz fixed point theorem, Banach fixed point theorem, Nielsen fixed point theorem and so forth. Besides, we refer some properties of the fixed point property of a digital k-retract.

• Honam Mathematical Journal
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• v.30 no.2
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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.

• Honam Mathematical Journal
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• v.36 no.1
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• pp.85-102
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• 2014

We show that if a graph G is compressed, then the proper part of the intersection poset of the corresponding graphical arrangement $A_G$ has the homotopy type of a wedge of spheres. Furthermore, we also indicate the number of spheres in the wedge, based on the number of adjacent edges of vertices in G.

• Journal of the Korean Mathematical Society
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• v.35 no.2
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• pp.371-386
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• 1998

Let G be a compact Lie group and M a semialgebraic G space in some orthogonal representation space of G. We prove that if G is finite then M has an equivariant semialgebraic triangulation. Moreover this triangulation is unique. When G is not finite we show that M has a semialgebraic G CW complex structure, and this structure is unique. As a consequence compact semialgebraic G space has an equivariant simple homotopy type.

• The Pure and Applied Mathematics
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• v.13 no.1 s.31
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• pp.11-18
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• 2006

Let f : (X, A) ${\rightarrow}$ (Y, B) be a map of pairs of compact polyhedra. A surplus Nielsen root number $SN(f;X\;{\backslash}\;A,\;c)$ is defined which is lower bound for the number of roots on X \ A for all maps in the homotopy class of f. It is shown that for many pairs this lower bound is the best possible one, as $SN(f;X\;{\backslash}\;A,\;c)$ can be realized without by-passing condition.

• Communications of the Korean Mathematical Society
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• v.11 no.4
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• pp.1077-1085
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• 1996

This paper provides examples which can not be approximate fibrations and shows that if $N^n$ is a closed aspherical manifold, $\pi_1(N)$ is hyperhophian, normally cohophian, and $\pi_1(N)$ has no nontrivial Abelian normal subgroup, then the product of $N^n$ and a sphre $S^m$ satisfies the property that all PL maps from an orientable manifold M to a polyhedron B for which each point preimage is homotopy equivalent to $N^n \times S^m$ necessarily are approximate fibrations.

• Journal of the Korean Mathematical Society
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• v.32 no.1
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• pp.33-40
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• 1995

A space Y is called finitely dominated if there is a finite complex K such that Y is a retract of K in the homotopy category, i.e., we require maps $i : Y \longrightarrow K and r : K \longrightarrow Y with r \circ i \simeq idy$. The following questions are very classical in topology.

• Communications of the Korean Mathematical Society
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• v.19 no.1
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• pp.159-167
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• 2004

In [1], a relative root Nielsen number $N_{rel}(f;\;c)$ is introduced which is a homotopy invariant lower bound for the number of roots at $c\;{\in}\;Y$ for a map of pairs of spaces $f\;:\;(X,\;A)\;{\rightarrow}\;(Y,\;B)$. In this paper, we obtain a minimum theorem for $N_{rel}(f;\;c)$ under some new assumptions on the spaces and maps which are different from those in [1].

• Communications of the Korean Mathematical Society
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• v.28 no.2
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• pp.361-369
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• 2013

We introduce a Nielsen type number of a fibre preserving map, and show that it is a lower bound for the number of $n$-orbits in the homotopy class. Under suitable conditions we show that it is equal to the Nielsen type relative essential $n$-orbit number. We also give necessary and sufficient conditions for it and the essential $n$-orbit number to coincide.

• Communications of the Korean Mathematical Society
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• v.11 no.2
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• pp.445-455
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• 1996

In this paper we study the Reidemeister number $R(f_G)$ for a self-map $f_G : (X, G) \to (X, G)$ of the transformation group (X,G), as an extenstion of the Reidemeister number R(f) for a self-map $f : X \to X$ of a topological space X.