• Bulletin of the Korean Mathematical Society
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• v.36 no.4
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• pp.755-762
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• 1999

Let X be a p-compace groyp, Y -> X bd a p-compact-subgroup of X and G -> X be a p-compact toral subgroup of X with $(X/Y)^{hG} \neq 0$. In this paper we show that the homotopy fixed point set of the homotopy fibre $(X/Y)^{hG}$ is $F_p$-finite.

• Communications of the Korean Mathematical Society
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• v.34 no.3
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• pp.991-1004
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• 2019

Firstly we consider preorders (not necessarily partial orders) on a canonical quotient of the set of the homotopy classes of continuous maps between two spaces induced by a certain equivalence relation ${\sim}_{{\varepsilon}R}$. Secondly we apply it to a classification of orientable fibrations over Y with fibre X. In the classification theorem of J. Stasheff [22] and G. Allaud [3], they use the set $[Y,\;Baut_1X]$ of homotopy classes of continuous maps from Y to $Baut_1X$, which is the classifying space for fibrations with fibre X due to A. Dold and R. Lashof [11]. In this paper we give a classification of fibrations using a preordered set (abbr., proset) structure induced by $[Y,\;Baut_1X]_{{\varepsilon}R}:=[Y,\;Baut_1X]/{\sim}_{{\varepsilon}R}$.

• Honam Mathematical Journal
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• v.29 no.1
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• pp.101-118
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• 2007

The notions of digital k-homotopy equivalence and digital ($k_0,k_1$)-homotopy equivalence were developed in [13, 16]. By the use of the digital k-homotopy equivalence, we can investigate digital k-homotopy equivalent properties of Cartesian products constructed by the minimal simple closed 4- and 8-curves in $\mathbf{Z}^2$.

• Bulletin of the Korean Mathematical Society
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• v.38 no.1
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• pp.143-148
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• 2001

First, we show the finiteness property of the homotopy fixed point set of p-discrete toral group. Let $G_\infty$ be a p-discrete toral group and X be a finite complex with an action of $G_\infty such that X^K$ is nilpotent for each finit p-subgroup K of $G_\infty$. Assume X is $F_\rho-complete$. Then X(sup)hG$\infty$ is F(sub)p-finite. Using this result, we give the condition so that X$^{hG}$ is $F_\rho-finite for \rho-compact$ toral group G.

• Journal of the Korean Mathematical Society
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• v.54 no.2
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• pp.399-415
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• 2017

Given a topological space X and a non-negative integer k, we study the self-homotopy equivalences of X that do not change maps from X to n-sphere $S^n$ homotopically by the composition for all $n{\geq}k$. We denote by ${\varepsilon}^{\sharp}_k(X)$ the set of all homotopy classes of such self-homotopy equivalences. This set is a dual concept of ${\varepsilon}^{\sharp}_k(X)$, which has been studied by several authors. We prove that if X is a finite CW complex, there are at most a finite number of distinguishing homotopy classes ${\varepsilon}^{\sharp}_k(X)$, whereas ${\varepsilon}^{\sharp}_k(X)$ may not be finite. Moreover, we obtain concrete computations of ${\varepsilon}^{\sharp}_k(X)$ to show that the cardinal of ${\varepsilon}^{\sharp}_k(X)$ is finite when X is either a Moore space or co-Moore space by using the self-closeness numbers.

• Bulletin of the Korean Mathematical Society
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• v.45 no.1
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• pp.67-73
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• 2008

The purpose of this paper is to present some fixed point results for nonself multivalued operators on a set with two metrics. In addition, a homotopy result for multivalued operators on a set with two metrics is given. The data dependence and the well-posedness of the fixed point problem are also discussed.

• Journal of the Korean Mathematical Society
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• v.56 no.5
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• pp.1371-1385
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• 2019

Given a topological space X and a non-negative integer k, ${\varepsilon}^{\sharp}_k(X)$ is the set of all self-homotopy equivalences of X that do not change maps from X to an t-sphere $S^t$ homotopically by the composition for all $t{\geq}k$. This set is a subgroup of the self-homotopy equivalence group ${\varepsilon}(X)$. We find certain homotopic tools for computations of ${\varepsilon}^{\sharp}_k(X)$. Using these results, we determine ${\varepsilon}^{\sharp}_k(M(G,n))$ for $k{\geq}n$, where M(G, n) is a Moore space type of (G, n) for a finitely generated abelian group G.

• Journal of the Korean Mathematical Society
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• v.47 no.5
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• pp.1031-1054
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• 2010

Let $\mathbb{Z}^n$ be the Cartesian product of the set of integers $\mathbb{Z}$ and let ($\mathbb{Z}$, T) and ($\mathbb{Z}^n$, $T^n$) be the Khalimsky line topology on $\mathbb{Z}$ and the Khalimsky product topology on $\mathbb{Z}^n$, respectively. Then for a set $X\;{\subset}\;\mathbb{Z}^n$, consider the subspace (X, $T^n_X$) induced from ($\mathbb{Z}^n$, $T^n$). Considering a k-adjacency on (X, $T^n_X$), we call it a (computer topological) space with k-adjacency and use the notation (X, k, $T^n_X$) := $X_{n,k}$. In this paper we introduce the notions of KD-($k_0$, $k_1$)-homotopy equivalence and KD-k-deformation retract and investigate a classification of (computer topological) spaces $X_{n,k}$ in terms of a KD-($k_0$, $k_1$)-homotopy equivalence.

• Bulletin of the Korean Mathematical Society
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• v.25 no.2
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• pp.175-178
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• 1988

Let .SIGMA. be a smooth compact manifold without boundary having the same homotopy type as a sphere, which is called a homotopy sphere. Supose a group G acts smoothly on .SIGMA. with the fixed point set .SIGMA.$^{G}$ consists of two isolated fixed points p and q. In this case, tangent spaces $T_{p}$ .SIGMA. and $T_{q}$ .SIGMA. at isolated fixed points, as isotropy representations of G are called Smith equivalent. Moreover .SIGMA. is called a supporting homotopy sphere of Smith equivalent representations $T_{p}$ .SIGMA. and $T_{q}$ .SIGMA.. The study on Smith equivalence has rich history, and for this we refer the reader to [P] or [Su]. The following question of pp.A.Smith [S] motivates the study on Smith equivalence.e.

• Korean Journal of Mathematics
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• v.18 no.2
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• pp.175-183
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• 2010

The classical group completion theorem states that under a certain condition the homology of ${\Omega}BM$ is computed by inverting ${\pi}_0M$ in the homology of M. McDuff and Segal extended this theorem in terms of homology fibration. Recently, more general group completion theorem for simplicial spaces was developed. In this paper, we construct a symmetric monoidal 2-category ${\mathcal{A}}$. The 1-morphisms of ${\mathcal{A}}$ are generated by three atomic 2-dimensional CW-complexes and the set of 2-morphisms is given by the group of path components of the space of homotopy equivalences of 1-morphisms. The main part of the paper is to compute the homotopy type of the group completion of the classifying space of ${\mathcal{A}}$, which is shown to be homotopy equivalent to ${\mathbb{Z}}{\times}BAut^+_{\infty}$.