• 대한수학회보
• /
• 제36권4호
• /
• pp.755-762
• /
• 1999

Let X be a p-compace groyp, Y -> X bd a p-compact-subgroup of X and G -> X be a p-compact toral subgroup of X with $(X/Y)^{hG} \neq 0$. In this paper we show that the homotopy fixed point set of the homotopy fibre $(X/Y)^{hG}$ is $F_p$-finite.

• 대한수학회논문집
• /
• 제11권1호
• /
• pp.235-244
• /
• 1996

For a self-map f of a CW-pair (X, A), we introduce the G(f)-sequence of (X, A) which consists of subgroups of homotopy groups in the homotopy sequence of (X, A) and show some properties of the relative homotopy Jian groups. We also show a condition for the G(f)-sequence to be exact.

• 대한수학회보
• /
• 제38권1호
• /
• pp.143-148
• /
• 2001

First, we show the finiteness property of the homotopy fixed point set of p-discrete toral group. Let $G_\infty$ be a p-discrete toral group and X be a finite complex with an action of $G_\infty such that X^K$ is nilpotent for each finit p-subgroup K of $G_\infty$. Assume X is $F_\rho-complete$. Then X(sup)hG$\infty$ is F(sub)p-finite. Using this result, we give the condition so that X$^{hG}$ is $F_\rho-finite for \rho-compact$ toral group G.

• 대한수학회지
• /
• 제43권3호
• /
• pp.491-506
• /
• 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.

• 충청수학회지
• /
• 제36권4호
• /
• pp.289-296
• /
• 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
• /
• 제56권5호
• /
• pp.1371-1385
• /
• 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
• /
• 제43권5호
• /
• pp.1047-1063
• /
• 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.

• 대한수학회지
• /
• 제45권4호
• /
• pp.1089-1100
• /
• 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.

• 대한수학회보
• /
• 제25권2호
• /
• pp.175-178
• /
• 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.

• 충청수학회지
• /
• 제6권1호
• /
• pp.113-130
• /
• 1993

In this paper, we introduce certain subgroups $E^{Rel}_n$ (X, Y, K, G) of the relative homotopy groups ${\sigma}_n$(X, Y, K, G) of a transformation group and examine the algebraic properties of these groups.