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

- Journal title : Journal of the Korean Mathematical Society
- Volume 43, Issue 5, 2006, pp.1047-1063
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2006.43.5.1047

Title & Authors

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

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

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

Abstract

Let be the subgroups of consisting of homotopy classes of self-homotopy equivalences that fix homotopy groups through the dimension of X and be the subgroup of that fix homology groups for all dimension. In this paper, we establish some connections between the homotopy group of X and the subgroup . We also give some relations between , as well as a generalized Gottlieb group , and a subset of [X, W]. Finally we establish a connection between the coGottlieb group of X and the subgroup of consisting of homotopy classes of self-homotopy equivalences that fix cohomology groups.

Keywords

self-homotopy equivalences;Gottlieb groups;coGottlieb groups;

Language

English

Cited by

References

1.

M. Arkowitz, G. Lupton, and A. Murillo, Subgroups of the group of self-homotopy equivalences, Contemporary Mathematics 274 (2001), 21-32

2.

M. Arkowitz and K. Maruyama, Self-homotopy equivalences which induce the identity on homology, cohomology or homotopy groups, Topology Appl. 87 (1998), no. 2, 133-154

3.

H. J. Baues, Homotopy type and homology, Clarendon Press, Oxford, 1996

4.

E. Dror and A. Zabrodsky, Unipotency and nilpotency in homotopy equivalences, Topology 18 (1979), no. 3, 187-197

6.

P. Hilton, Homotopy theory and duality, Gordon and Breach Science Publishers, New York-London Paris, 1965

8.

K. Maruyama, Localization of a certain subgroup of self-homotopy equivalences, Pacific J. Math. 136 (1989), no. 2, 293-301

9.

K. Maruyama, Localization of self-homotopy equivalences inducing the identity on ho- mology, Math. Proc. Cambridge Philos. Soc. 108 (1990), no. 2, 291-297

10.

11.

N. Oda, Pairings and copairings in the category of topological spaces, Publ. Res. Inst. Math. Sci. 28 (1992), no. 1, 83-97

12.

S. Oka, N. Sawashita, and M. Sugawara, On the group of self-equivalences of a mapping cone, Hiroshima Math. J. 4 (1974), 9-28

13.

J. W. Rutter, A homotopy classi¯cation of maps into an induced fibre space, Topology 6 (1967), 379-403

14.

K. Varadarajan, Generalised Gottlieb groups, J. Indian Math. Soc. 33 (1969), 141-164

15.

G. W. Whitehead, Elements of homotopy theory, Graduate texts in Mathematics 61, Springer-Verlag, New York Heidelberg Berlin, 1978

16.

M. H. Woo and J.-R. Kim, Certain subgroups and homotopy groups, J. Korean. Math. Soc. 21 (1984), no. 2, 109-120