• 충청수학회지
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• 제14권1호
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• pp.1-6
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• 2001

We investigate the relationships between the Gottlieb groups and the generalized Gottlieb groups, and study some properties of the generalized Gottlieb groups. Lee and Woo [5] proved that $G_n(X,i_1,X{\times}Y){\simeq_-}G_n(X){\oplus}{\pi}_n(Y)$. We can easily re-prove the above main theorem of [5] using some properties of the generalized Gottlieb groups, and obtain a more powerful result as follows; if $F{\rightarrow}^iE{\rightarrow}^pB$ is a homotopically trivial fibration, then $G_n(F,i,E){\simeq_-}{\pi}_n(B){\oplus}G_n(F)$.

• 충청수학회지
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• 제9권1호
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• pp.77-87
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• 1996

For any map $v:X{\rightarrow}Y$, the generalized Gottlieb set $G({\Sigma}A;X,v,Y)$ with respect to v is a subgroup of $[{\Sigma}A,Y]$. If $v:X{\rightarrow}Y$ has a left homotopy inverse $u:X{\rightarrow}Y$, then for any $f{\in}G({\Sigma}A;X,v,Y)$, $g{\in}G({\Sigma}A;X,u,Y)$, the function spaces $L({\Sigma}A,X;uf)$ and $L({\Sigma}A,X;g)$ have the same homotopy type.

• 충청수학회지
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• 제20권3호
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• pp.245-259
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• 2007

We define and study a concept of $T^f$-space for a map, which is a generalized one of a T-space, in terms of the Gottlieb set for a map. We show that X is a $T_f$-space if and only if $G({\Sigma}B;A,f,X)=[{\Sigma}B,X]$ for any space B. For a principal fibration $E_k{\rightarrow}X$ induced by $k:X{\rightarrow}X^{\prime}$ from ${\epsilon}:PX^{\prime}{\rightarrow}X^{\prime}$, we obtain a sufficient condition to having a lifting $T^{\bar{f}}$-structure on $E_k$ of a $T^f$-structure on X. Also, we define and study a concept of co-$T^g$-space for a map, which is a dual one of $T^f$-space for a map. We obtain a dual result for a principal cofibration $i_r:X{\rightarrow}C_r$ induced by $r:X^{\prime}{\rightarrow}X$ from ${\iota}:X^{\prime}{\rightarrow}cX^{\prime}$.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제14권4호
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• pp.289-306
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• 2007

We define and study a concept of $H^f-space$ for a map, which is a generalized concept of an H-space, in terms of the Gottlieb set for a map. For a principal fibration $E_{\kappa}{\rightarrow}X$ induced by ${\kappa}:X{\rightarrow}X'\;from\;{\epsilon}:\;PX'{\rightarrow}X'$, we can obtain a sufficient condition to having an $H^{\bar{f}}-structure\;on\;E_{\kappa}$, which is a generalization of Stasheff's result [17]. Also, we define and study a concept of $co-H^g-space$ for a map, which is a dual concept of $H^f-space$ for a map. Also, we get a dual result which is a generalization of Hilton, Mislin and Roitberg's result [6].