• Communications of the Korean Mathematical Society
• /
• v.11 no.1
• /
• 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.

• Communications of the Korean Mathematical Society
• /
• v.33 no.1
• /
• pp.53-71
• /
• 2018

In this paper, we provide two different descriptions for a relative derived category with respect to a subcategory ${\mathcal{X}}$ of an abelian category ${\mathcal{A}}$. First, we construct an exact model structure on certain exact category which has as its homotopy category the relative derived category of ${\mathcal{A}}$. We also show that a relative derived category is equivalent to homotopy category of certain complexes. Moreover, we investigate the existence of certain recollements in such categories.

• 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.

• 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 Korean Mathematical Society
• /
• v.33 no.4
• /
• pp.709-716
• /
• 1996

Nielsen coincidence theory is concerned with the determinatin of a lower bound of the minimal number MC[f,g] of coincidence points for all maps in the homotopy class of a given map (f,g) : X $\to$ Y. The Nielsen Nielsen number $N_R(f,g)$ (similar to [9]) is introduced in [3], which is a lower bound for the number of coincidence points in the relative homotopy class of (f,g) and $N_R(f,g) \geq N(f,g)$.

• Journal of the Korean Mathematical Society
• /
• v.44 no.6
• /
• pp.1479-1503
• /
• 2007

In this paper, we study a strong k-deformation retract derived from a relative k-homotopy and investigate its properties in relation to both a k-homotopic thinning and the k-fundamental group. Moreover, we show that the k-fundamental group of a wedge product of closed k-curves not k-contractible is a free group by the use of some properties of both a strong k-deformation retract and a digital covering. Finally, we write an algorithm for calculating the k-fundamental group of a dosed k-curve by the use of a k-homotopic thinning.

• Communications of the Korean Mathematical Society
• /
• v.7 no.1
• /
• pp.15-24
• /
• 1992

• Communications of the Korean Mathematical Society
• /
• v.12 no.3
• /
• pp.709-715
• /
• 1997

For a fibration (E,B,p) with fiber F and a fiber map f, we show that if the inclusion $i : F \to E$ has a left homotopy inverse, then $G^f_n(E,F)$ is isomorphic to $G^f_n(F,E) \oplus \pi_n(B)$. In particular, by taking f as the identity map on E we have $G_n(E,F)$ is isomorphic to $G_n(F) \oplus \pi_n(B)$.

• Journal of the Korean Mathematical Society
• /
• v.32 no.2
• /
• pp.171-181
• /
• 1995

Nielsen coincidence theory is concerned with the estimation of a lower bound for the number of coincidences of two maps $f,g: X \longrightarrow Y$. For this purpose the so-called Nielsen number N(f,g) is introduced, which is a lower bound for the number of coincidences ([1]). The relative Nielsen number N(f : X,A) in the fixed point theory is introduced in [3], which is a lower bound for the number of fixed points for all maps in the relative homotopy class of f:(X,A) $\longrightarrow$ (X,A), and its estimation is given in [5].

• Communications of the Korean Mathematical Society
• /
• v.11 no.1
• /
• pp.245-252
• /
• 1996

The relative Nielsen number N(f;X,A) was introduced in 1986. It gives us a better, and ideally sharp, lower bound for the minimum number MF[f;X,A] of fixed points in the homotopy class of the map $f;(X,A) \to (X,A)$. Similarly, we also can think about the Nielsen map $f:(X,A) \to (X,A)$. Similarly, we also can be think about the Nielsen root theory. In this paper, we introduce a relative root Nielsen number N(f;X,A,c) of $f:(X,A) \to (Y,B)$ and show some basic properties.