Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지
DOI QR코드

DOI QR Code

G'p-SPACES FOR MAPS AND HOMOLOGY DECOMPOSITIONS

  • Yoon, Yeon Soo (Department of Mathematics Education Hannam University)
  • Received : 2015.09.16
  • Accepted : 2015.10.26
  • Published : 2015.11.15
Download PDF
⟨ Previous Next ⟩

Abstract

For a map $p:X{\rightarrow}A$, we define and study a concept of $G^{\prime}_p$-space for a map, which is a generalized one of a G'-space. Any G'-space is a $G^{\prime}_p$-space, but the converse does not hold. In fact, $CP^2$ is a $G^{\prime}_{\delta}$-space, but not a G'-space. It is shown that X is a $G^{\prime}_p$-space if and only if $G^n(X,p,A)=H^n(X)$ for all n. We also obtain some results about $G^{\prime}_p$-spaces and homology decompositions for spaces. As a corollary, we can obtain a dual result of Haslam's result about G-spaces and Postnikov systems.

Keywords

Acknowledgement

Supported by : Hannam University

References

  1. J. Aguade, Decomposable free loop spaces, Canad. J. Math. 39 (1987), 938-955. https://doi.org/10.4153/CJM-1987-047-9
  2. B. Eckmann and P. Hilton, Decomposition homologique d'un polyedre simplement connexe, Canad. J. Math. 248 (1959), 2054-2558.
  3. D. H. Gottlieb, A certain subgroup of the fundamental group, Amer. J. Math. 87 (1965), 840-856. https://doi.org/10.2307/2373248
  4. D. H. Gottlieb, Evaluation subgroups of homotopy groups, Amer. J. Math. 91 (1969), 729-756. https://doi.org/10.2307/2373349
  5. Marek Golansinski and John R. Klein, On maps into a co-H-space, Hiroshima Math. J. 28 (1998), 321-327.
  6. H. B. Haslam, G-spaces and H-spaces, Ph. D. Thesis, Univ. of California, Irvine, 1969.
  7. P. Hilton, Homotopy Theory and Duality, Gordon and Breach Science Pub., 1965.
  8. D. W. Kahn, Induced maps for Postnikov systems, Trans. Amer. Math. Soc. 107 (1963), 432-450.
  9. D. W. Kahn, A note on H-spaces and Postnikov systems of spheres, Proc. Amer. Math. Soc. 15 (1964), 300-307.
  10. K. L. Lim, On cyclic maps, J. Austral. Math. Soc. (Series A) 32 (1982), 349-357. https://doi.org/10.1017/S1446788700024903
  11. K. L. Lim, Cocyclic maps and coevaluation subgroups, Canad. Math. Bull. 30 (1987), 63-71. https://doi.org/10.4153/CMB-1987-009-1
  12. Christopher McCord and John Oprea, Rational Ljusternik-Schnirelmann Category and the Arnol'd conjecture for nilmanifolds, Topology 32 (1993), no. 4, 701-717. https://doi.org/10.1016/0040-9383(93)90047-Y
  13. N. Oda, The homotopy of the axes of pairings , Canad. J. Math. 17 (1990), 856-868.
  14. Graham Hilton Toomer, Liusternik-Schnirelmann Category and the Moore spectral sequence, Ph. D. dissertation, Cornell University, 1974.
  15. Graham Hilton Toomer, Two applications of homology decompositions, Can. J. Math. 27 (1975), no. 2, 323-329. https://doi.org/10.4153/CJM-1975-039-1
  16. K. Varadarajan, Genralized Gottlieb groups, J. Indian Math. Soc. 33 (1969), 141-164.
  17. G. W. Whitehead, Elements of homotopy theory, Springer-Verlag, New York Inc., 1978.
  18. M. H. Woo and Y. S. Yoon, T-spaces by the Gottlieb groups and duality, J. Austral. Math. Soc. (Series A) 59 (1995), 193-203. https://doi.org/10.1017/S1446788700038593
  19. Y. S. Yoon, Lifting Gottlieb sets and duality, Proc. Amer. Math. Soc. 119 (1993), no. 4, 1315-1321. https://doi.org/10.1090/S0002-9939-1993-1184089-7
  20. Y. S. Yoon, The generalized dual Gottlieb sets, Topology Appl. 109 (2001), 173-181. https://doi.org/10.1016/S0166-8641(99)00150-9
  21. Y. S. Yoon, $G^f$-spaces for maps and Postnikov systems, J. Chungcheong Math. Soc. 22 (2009), no. 4, 831-841.
  22. Y. S. Yoon, $H^f$-spaces for maps and thier duals, J. Korean Soc. Math. Educ. Ser. B. 14 (2007), no. 4, 289-306.
  23. Y. S. Yoon, lifting T-structures and their duals, J. Chungcheong Math. Soc. 20 (2007), no. 3, 245-259.

Cited by

  1. PRINCIPAL COFIBRATIONS AND GENERALIZED CO-H-SPACES vol.30, pp.1, 2017, https://doi.org/10.14403/jcms.2017.30.1.129