• Bulletin of the Korean Mathematical Society
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• v.41 no.3
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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.

• Bulletin of the Korean Mathematical Society
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• v.36 no.4
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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.

• Communications of the Korean Mathematical Society
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• v.25 no.1
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• pp.111-117
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• 2010

For a based, 1-connected, finite CW-complex X, we denote by $\varepsilon(X)$ the group of homotopy classes of self-homotopy equivalences of X and by $\varepsilon_#\;^{dim+r}(X)$ the subgroup of homotopy classes which induce the identity on the homotopy groups of X in dimensions $\leq$ dim X+r. In this paper, we calculate the subgroups $\varepsilon_#\;^{dim+r}(X)$ when X is a wedge of two Moore spaces determined by cyclic groups and in consecutive dimensions.

• Journal of the Korean Mathematical Society
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• v.43 no.5
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• pp.1047-1063
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• 2006

Let $\varepsilon_#(X)$ be the subgroups of $\varepsilon(X)$ consisting of homotopy classes of self-homotopy equivalences that fix homotopy groups through the dimension of X and $\varepsilon_*(X) $ be the subgroup of $\varepsilon(X)$ that fix homology groups for all dimension. In this paper, we establish some connections between the homotopy group of X and the subgroup $\varepsilon_#(X)\cap\varepsilon_*(X)\;of\;\varepsilon(X)$. We also give some relations between $\pi_n(W)$, as well as a generalized Gottlieb group $G_n^f(W,X)$, and a subset $M_{#N}^f(X,W)$ of [X, W]. Finally we establish a connection between the coGottlieb group of X and the subgroup of $\varepsilon(X)$ consisting of homotopy classes of self-homotopy equivalences that fix cohomology groups.

• Journal of the Korean Mathematical Society
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• v.61 no.3
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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.

• 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.

• Journal of the Korean Mathematical Society
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• v.43 no.3
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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.

• Journal of the Korean Mathematical Society
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• v.45 no.4
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• pp.1089-1100
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• 2008

For a pointed space X, the subgroups of self-homotopy equivalences $Aut_{\sharp}_N(X)$, $Aut_{\Omega}(X)$, $Aut_*(X)$ and $Aut_{\Sigma}(X)$ are considered, where $Aut_{\sharp}_N(X)$ is the group of all self-homotopy classes f of X such that $f_{\sharp}=id\;:\;{\pi_i}(X){\rightarrow}{\pi_i}(X)$ for all $i{\leq}N{\leq}{\infty}$, $Aut_{\Omega}(X)$ is the group of all the above f such that ${\Omega}f=id;\;Aut_*(X)$ is the group of all self-homotopy classes g of X such that $g_*=id\;:\;H_i(X){\rightarrow}H_i(X)$ for all $i{\leq}{\infty}$, $Aut_{\Sigma}(X)$ is the group of all the above g such that ${\Sigma}g=id$. We will prove that $Aut_{\Omega}(X_1{\times}\cdots{\times}X_n)$ has two factorizations similar to those of $Aut_{\sharp}_N(X_1{\times}\cdots{\times}\;X_n)$ in reference [10], and that $Aut_{\Sigma}(X_1{\vee}\cdots{\vee}X_n)$, $Aut_*(X_1{\vee}\cdots{\vee}X_n)$ also have factorizations being dual to the former two cases respectively.

• 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.

• 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.