• Journal of the Korean Mathematical Society
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• v.44 no.6
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• pp.1479-1503
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• 2007

In this paper, we study a strong k-deformation retract derived from a relative k-homotopy and investigate its properties in relation to both a k-homotopic thinning and the k-fundamental group. Moreover, we show that the k-fundamental group of a wedge product of closed k-curves not k-contractible is a free group by the use of some properties of both a strong k-deformation retract and a digital covering. Finally, we write an algorithm for calculating the k-fundamental group of a dosed k-curve by the use of a k-homotopic thinning.

• 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.

• East Asian mathematical journal
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• v.19 no.1
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• pp.113-120
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• 2003

We study how the second homotopy modules of group presentations are transformed by Tietze transformations and discuss some application.

• Journal of the Korean Mathematical Society
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• v.16 no.1
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• pp.87-93
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• 1979

In this note the geometric realization of a finite group of homotopy classes of self homotopy equivalences by a finite group of diffeomorphisms is investigated. In order for this to be accomplished an algebraic condition, which guarantees a certain group extension exists, must be satisfied. It is shown for a geometrically interesting class of aspherical manifolds, called injective Seifert fiber spaces over a locally symmetric space, that this necessary algebraic condition is also sufficient for geometric realization.

• Journal of the Korean Mathematical Society
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• v.34 no.3
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• pp.653-659
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• 1997

In this paper, we extend the Gottlieb groups of a space to the Gottlieb groups of a map and show some properties of those groups. Especially, We show the 2nd Gottlieb group of a map is contained in the center of the homotopy group of the map and show $G_n(F) = \pi_n(f)$ for an H-map f between H-spaces. We also show the Gottlieb subgroups $G_n(A), G_n(X) and G_n(f)$ make a sequence if the map $f : A \to X$ has a right homotopy inverse.

• Bulletin of the Korean Mathematical Society
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• v.45 no.1
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• pp.119-131
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• 2008

In this paper, we generalize the Gottlieb groups and the related G-sequence of those groups, and present some sufficient conditions to ensure the exactness or non-exactness of G-sequences at some terms. We also give some applications of the exactness or non-exactness of G-sequences. Especially, we show that the non-exactness of G-sequences implies the non-triviality of homotopy groups of some function spaces.

• The Pure and Applied Mathematics
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• v.16 no.2
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• pp.193-198
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• 2009

In the previous work [5] we have determined the group ${{\varepsilon}_{\sharp}}^{dim+r}^{dim+r}(X)$ for $X\;=\;M(Z_q,\;n+1){\vee}M(Z_q,\;n)$ for all integers q > 1. In this paper, we investigate the group ${{\varepsilon}_{\sharp}}^{dim+r}(X)$ for $X\;=\;M(Z{\oplus}Z_q,\;n+1){\vee}M(Z{\oplus}Z_q,\;n)$ for all odd numbers q > 1.

• Korean Journal of Mathematics
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• v.6 no.2
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• pp.165-172
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• 1998

For a G-map ${\phi}:X{\rightarrow}X$, we define and characterize the Reidemeister number $R_G({\phi})$ of ${\phi}$. Also, we prove that $R_G({\phi})$ is a G-homotopy invariance and we obtain a lower bound of $R_G({\phi})$.

• Journal of the Korean Mathematical Society
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• v.54 no.4
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• pp.1231-1242
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• 2017

In this paper, we show that, under some conditions, self-maps of product spaces can be represented by the composition of two specific self-maps if their induced homomorphism on the i-th homotopy group is an automorphism for all i in some section of positive integers. As an application, we obtain closeness numbers of several product spaces.

• Korean Journal of Mathematics
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• v.26 no.3
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• pp.439-458
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• 2018

In this study, we interpret the notion of homotopy of morphisms in the category of crossed modules in a category C of groups with operations using the categorical equivalence between the categories of crossed modules and of internal categories in C. Further, we characterize the derivations of crossed modules in a category C and obtain new crossed modules using regular derivations of old one.