• 대한수학회보
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• 제36권4호
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• pp.649-654
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• 1999

We consider the forgetful map from the group of equivariant self equivalences to the group of non-equivariant self equivalences. A sufficient condition for this forgetful map being a monomorphism is obtained. Several examples are given.

• 대한수학회보
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• 제41권3호
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• pp.393-401
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• 2004

In this paper, we apply the concept of the group \ulcorner(X,A) of self pair homotopy equivalences of a CW-pair (X, A) to the Postnikov system. By using a short exact sequence related to the group of self pair homotopy equivalences, we obtain the following result: for any Postnikov section X$\sub$n/ of a CW-complex X, the group \ulcorner(X$\sub$n/, A) of self pair homotopy equivalences on the pair (X$\sub$n/, X) is isomorphic to the group \ulcorner(X) of self homotopy equivalences on X. As a corollary, we have, \ulcorner(K($\pi$, n), M($\pi$, n)) ≡ \ulcorner(M($\pi$, n)) for each n$\pi$1, where K($\pi$,n) is an Eilenberg-Mclane space and M($\pi$,n) is a Moore space.

• 충청수학회지
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• 제36권4호
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• pp.289-296
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• 2023

When G is an abelian group, we use the notation M(G, n) to denote the Moore space. The space X is the wedge product space of Moore spaces, given by X = M(ℤ_q, n+ 2) ∨ M(ℤ_q, n+ 1) ∨ M(ℤ_q, n). We determine the self-homotopy classes group [X, X] and the self-homotopy equivalence group 𝓔(X). We investigate the subgroups of [M_j , M_k] consisting of homotopy classes of maps that induce the trivial homomorphism up to (n + 2)-homotopy groups for j ≠ k. Using these results, we calculate the subgroup 𝓔^dim_#(X) of 𝓔(X) in which all elements induce the identity homomorphism up to (n + 2)-homotopy groups of X.

• 대한수학회지
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• 제43권3호
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• pp.491-506
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• 2006

In this paper, we extend the concept of the group ${\varepsilon}(X)$ of self homotopy equivalences of a space X to that of an object in the category of pairs. Mainly, we study the group ${\varepsilon}(X,\;A)$ of pair homotopy equivalences from a CW-pair (X, A) to itself which is the special case of the extended concept. For a CW-pair (X, A), we find an exact sequence $1\;{\to}\;G\;{\to}\;{\varepsilon}(X,\;A)\;{to}\;{\varepsilon}(A)$ where G is a subgroup of ${\varepsilon}(X,\;A)$. Especially, for CW homotopy associative and inversive H-spaces X and Y, we obtain a split short exact sequence $1\;{\to}\;{\varepsilon}(X)\;{\to}\;{\varepsilon}(X{\times}Y,Y)\;{\to}\;{\varepsilon}(Y)\;{\to}\;1$ provided the two sets $[X{\wedge}Y,\;X{\times}Y]$ and [X, Y] are trivial.

• 대한수학회지
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• 제56권5호
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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.

• 대한수학회지
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• 제54권2호
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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.

• 대한수학회지
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• 제61권3호
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• pp.409-425
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• 2024

The category of pairs is the category whose objects are maps between two based spaces and morphisms are pair-maps from one object to another object. To study the self-homotopy equivalences in the category of pairs, we use covariant functors from the category of pairs to the group category whose objects are groups and morphisms are group homomorphisms. We introduce specific subgroups of groups of self-pair homotopy equivalences and put these groups together into certain sequences. We investigate properties of these sequences, in particular, the exactness and split. We apply the results to two special functors, homotopy and homology functors and determine the suggested several subgroups of groups of self-pair homotopy equivalences.

• 대한수학회지
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• 제54권4호
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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.

• 대한수학회보
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• 제58권2호
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• pp.445-449
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• 2021

We define the relative self-closeness number N��(g) of a map g : X → Y, which is a generalization of the self-closeness number N��(X) of a connected CW complex X defined by Choi and Lee [1]. Then we compare N��(p) with N��(X) for a fibration $X{\rightarrow}E{\rightarrow\limits^p}Y$. Furthermore we obtain its rationalized result.

• 대한수학회논문집
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• 제9권4호
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• pp.933-944
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• 1994

Let $\pi : P \to S^4$ be a principal G-bundle over $S^4$ whose the structure group G is a compact, connected, simple Lie group. Since $\pi_3(G) = \pi_4 (BG) = Z$, we can classify the principal bundle $P_k$ over $S^4$ by the map $S^4 \to BG$ of degree k. Atiyah and Jones [2] showed that $C_k = A_k/g^b_k$ is homotopy equivalent to $\Omega^3_k G \simeq \Omega^4_k BG$ where $A_k$ is the space of the all connections in $P_k$ and $g^b_k$ is the based gauge group which consists of all base point preserving automorphisms on $P_k$. Here $\Omega^nX$ is the space of all base-point preserving continuous map from $S^n$ to X. Let $M_k$ be the space of based gauge equivalence classes of all connections in $P_k$ satisfying the Yang-Mills self-duality equations, which we call the moduli space of G instantons.