• Title/Summary/Keyword: coGottlieb group

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THE GENERALIZED COGOTTLIEB GROUPS, RELATED ACTIONS AND EXACT SEQUENCES

  • Choi, Ho-Won;Kim, Jae-Ryong;Oda, Nobuyuki
    • Journal of the Korean Mathematical Society
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    • v.54 no.5
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    • pp.1623-1639
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    • 2017
  • The generalized coGottlieb sets are not known to be groups in general. We study some conditions which make them groups. Moreover, there are actions on the generalized coGottlieb sets which are different from known actions up to now. We give related exact sequence of the generalized coGottlieb sets. Using them, we obtain certain results related to the maps which preserve generalized coGottlieb sets.

GOTTLIEB GROUPS AND SUBGROUPS OF THE GROUP OF SELF-HOMOTOPY EQUIVALENCES

  • Kim, Jae-Ryong;Oda, Nobuyuki;Pan, Jianzhong;Woo, Moo-Ha
    • 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.

GENERALIZED T-SPACES AND DUALITY

  • YOON, YEON SOO
    • Honam Mathematical Journal
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    • v.27 no.1
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    • pp.101-113
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    • 2005
  • We define and study a concept of $T_A$-space which is closely related to the generalized Gottlieb group. We know that X is a $T_A$-space if and only if there is a map $r:L(A,\;X){\rightarrow}L_0(A,\;X)$ called a $T_A$-structure such that $ri{\sim}1_{L_0(A,\;X)}$. The concepts of $T_{{\Sigma}B}$-spaces are preserved by retraction and product. We also introduce and study a dual concept of $T_A$-space.

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