DOI QR코드

DOI QR Code

GOTTLIEB SUBSETS WITH RESPECT TO A MORPHISM IN THE CATEGORY OF PAIRS

  • Kim, Ji-Yean (DEPARTMENT OF MATHEMATICS KOREA UNIVERSITY) ;
  • Lee, Kee-Young (DEPARTMENT OF INFORMATION AND MATHEMATICS KOREA UNIVERSITY)
  • Received : 2009.05.08
  • Published : 2010.11.30

Abstract

We introduce the concept of cyclic morphisms with respect to a morphism in the category of pairs as a generalization of the concept of cyclic maps and we use the concept to obtain certain sets of homotopy classes in the category of pairs. For these sets, we get complete or partial answers to the following questions: (1) Is the concept the most general concept in the class of all concepts of generalized Gottlieb subsets introduced by many authors until now? (2) Are they homotopy invariants in the category of pairs? (3) When do they have a group structure?.

Keywords

References

  1. D. H. Gottlieb, A certain subgroup of the fundamental group, Amer. J. Math. 87 (1965), 840-856. https://doi.org/10.2307/2373248
  2. D. H. Gottlieb, Evaluation subgroups of homotopy groups, Amer. J. Math. 91 (1969), 729-756. https://doi.org/10.2307/2373349
  3. D. H. Gottlieb, Covering transformations and universal fibrations, Illinois J. Math. 13 (1969), 432-437.
  4. P. J. Hilton, Homotopy Theory and Duality, Mimeographed Notes, Cornell Univ. Ithaca, NY, 1959.
  5. G. E. Lang, Evaluation subgroups of factor spaces, Pacific J. Math. 42 (1972), 701-709. https://doi.org/10.2140/pjm.1972.42.701
  6. K. Y. Lee and M. H. Woo, The G-sequence and the ${\omega}-homology$ of a CW-pair, Topology Appl. 52 (1993), no. 3, 221-236. https://doi.org/10.1016/0166-8641(93)90104-L
  7. K. Y. Lee and M. H. Woo, Cyclic morphisms in the category of pairs and generalized G-sequences, J. Math. Kyoto Univ. 38 (1998), no. 2, 271-285. https://doi.org/10.1215/kjm/1250518118
  8. K. Y. Lee, M. H. Woo, and X. Zhao, Certain generalizations of G-sequences and their exactness, Bull. Korean Math. Soc. 45 (2008), no. 1, 119-131. https://doi.org/10.4134/BKMS.2008.45.1.119
  9. K. L. Lim, On cyclic maps, J. Austral. Math. Soc. Ser. A 32 (1982), no. 3, 349-357. https://doi.org/10.1017/S1446788700024903
  10. N. Oda, The homotopy set of the axes of pairings, Canad. J. Math. 42 (1990), no. 5, 856-868. https://doi.org/10.4153/CJM-1990-044-3
  11. J. Pan, X. Shen, and M. Woo, The G-sequence of a map and its exactness, J. Korean Math. Soc. 35 (1998), no. 2, 281-294.
  12. J. Siegel, G-spaces, W-spaces and H-spaces, Pacific J. Math. 31 (1969), 209-214. https://doi.org/10.2140/pjm.1969.31.209
  13. K. Varadarajian, Generalised Gottlieb groups, J. Indian Math. Soc. (N.S.) 33 (1969), 141-164.
  14. M. Woo and J. Kim, Certain subgroups of homotopy groups, J. Korean Math. Soc. 21 (1984), no. 2, 109-120.
  15. M. H. Woo and K. Y. Lee, On the relative evaluation subgroups of a CW-pair, J. Korean Math. Soc. 25 (1988), no. 1, 149-160. https://doi.org/10.4134/CKMS.2010.25.1.149