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

• Journal of the Korean Mathematical Society
• /
• v.45 no.4
• /
• 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.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.3 no.2
• /
• pp.37-49
• /
• 1999

In this paper we develop a general Homotopy method called the Group Homotopy method to solve the symmetric eigenproblem. The Group Homotopy method overcomes notable drawbacks of the existing Homotopy method, namely, (i) the possibility of breakdown or having a slow rate of convergence in the presence of clustering of the eigenvalues and (ii) the absence of any definite criterion to choose a step size that guarantees the convergence of the method. On the other hand, We also have a good approximations of the largest eigenvalue of a Symmetric matrix from Lanczos algorithm. We apply it for the largest eigenproblem of a very large symmetric matrix with a good initial points.

• Journal of the Korean Mathematical Society
• /
• v.56 no.5
• /
• 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.

• Korean Journal of Mathematics
• /
• v.5 no.2
• /
• pp.107-112
• /
• 1997

For a ring R with 1, the higher K-theory of Quillen is defined by the higher homotopy groups of the plus construction of the general linear group of R. On the other hand, the Volodin K-theory is defined by the higher homotopy groups of the Volodin space. In this paper we show that these two K-theories are equivalent. We show that the Volodin space is a homotopy fiber of the acyclic map from BGL(R) to its plus construction.

• Honam Mathematical Journal
• /
• v.36 no.3
• /
• pp.519-530
• /
• 2014

To study a deformation of a digital space from the viewpoint of digital homotopy theory, we have often used the notions of a weak k-deformation retract [20] and a strong k-deformation retract [10, 12, 13]. Thus the papers [10, 12, 13, 16] firstly developed the notion of a strong k-deformation retract which can play an important role in studying a homotopic thinning of a digital space. Besides, the paper [3] deals with a k-deformation retract and its homotopic property related to a digital fundamental group. Thus, as a survey article, comparing among a k-deformation retract in [3], a strong k-deformation retract in [10, 12, 13], a weak deformation k-retract in [20] and a digital k-homotopy equivalence [5, 24], we observe some relationships among them from the viewpoint of digital homotopy theory. Furthermore, the present paper deals with some parts of the preprint [10] which were not published in a journal (see Proposition 3.1). Finally, the present paper corrects Boxer's paper [3] as follows: even though the paper [3] referred to the notion of a digital homotopy equivalence (or a same k-homotopy type) which is a special kind of a k-deformation retract, we need to point out that the notion was already developed in [5] instead of [3] and further corrects the proof of Theorem 4.5 of Boxer's paper [3] (see the proof of Theorem 4.1 in the present paper). While the paper [4] refers some properties of a deck transformation group (or an automorphism group) of digital covering space without any citation, the study was early done by Han in his paper (see the paper [14]).

• Journal of the Korean Mathematical Society
• /
• v.54 no.2
• /
• pp.399-415
• /
• 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.

• Communications of the Korean Mathematical Society
• /
• v.12 no.1
• /
• pp.75-78
• /
• 1997

We show that if a simple $C^*$-algebra A satisfies certain $K_1$-group conditions, then two unitarily equivalent projections are homotopic. Also we show that the equivalence of projections determined by a dimension function is a homotopy.

• Journal of the Chungcheong Mathematical Society
• /
• v.6 no.1
• /
• 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.

• Journal of the Chungcheong Mathematical Society
• /
• v.5 no.1
• /
• pp.9-21
• /
• 1992

We define a certain subgroup $E_n$ (X, K, G) of the relative homotopy groups of a transformation group ${\sigma}_n$(X, K, G). The algebraic properties of these groups are examined.