Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

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

  • Kim, Jae-Ryong (Department of Mathematics Kookmin University) ;
  • Oda, Nobuyuki (Department of Applied Mathematics Fukuoka University) ;
  • Pan, Jianzhong (Institute of Mathematics Academy Sciences of Ching) ;
  • Woo, Moo-Ha (Department of Mathematics Education Korea University)
  • Published : 2006.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

Let $\varepsilon_#(X)$ be the subgroups of $\varepsilon(X)$ consisting of homotopy classes of self-homotopy equivalences that fix homotopy groups through the dimension of X and $\varepsilon_*(X) $ be the subgroup of $\varepsilon(X)$ that fix homology groups for all dimension. In this paper, we establish some connections between the homotopy group of X and the subgroup $\varepsilon_#(X)\cap\varepsilon_*(X)\;of\;\varepsilon(X)$. We also give some relations between $\pi_n(W)$, as well as a generalized Gottlieb group $G_n^f(W,X)$, and a subset $M_{#N}^f(X,W)$ of [X, W]. Finally we establish a connection between the coGottlieb group of X and the subgroup of $\varepsilon(X)$ consisting of homotopy classes of self-homotopy equivalences that fix cohomology groups.

Keywords

References

  1. M. Arkowitz, G. Lupton, and A. Murillo, Subgroups of the group of self-homotopy equivalences, Contemporary Mathematics 274 (2001), 21-32 https://doi.org/10.1090/conm/274/04453
  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 https://doi.org/10.1016/S0166-8641(97)00162-4
  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 https://doi.org/10.1016/0040-9383(79)90002-8
  5. D. H. Gottlieb, Evaluation subgroups of homotopy groups, Amer. J. Math. 91 (1969), 729-756 https://doi.org/10.2307/2373349
  6. P. Hilton, Homotopy theory and duality, Gordon and Breach Science Publishers, New York-London Paris, 1965
  7. P. J. Kahn, Self-equivalences of (n-1)-connected 2n-manifolds, Math. Ann. 180 (1969), 26-47 https://doi.org/10.1007/BF01350084
  8. K. Maruyama, Localization of a certain subgroup of self-homotopy equivalences, Pacific J. Math. 136 (1989), no. 2, 293-301 https://doi.org/10.2140/pjm.1989.136.293
  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 https://doi.org/10.1017/S0305004100069140
  10. K. Maruyama, Stability properties of maps between Hopf spaces, Q. J. Math. 53 (2002), no. 1, 47-57 https://doi.org/10.1093/qjmath/53.1.47
  11. N. Oda, Pairings and copairings in the category of topological spaces, Publ. Res. Inst. Math. Sci. 28 (1992), no. 1, 83-97 https://doi.org/10.2977/prims/1195168857
  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 https://doi.org/10.1016/0040-9383(67)90025-0
  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

Cited by

  1. Cocyclic element preserving pair maps and fibrations vol.191, 2015, https://doi.org/10.1016/j.topol.2015.05.052
  2. The set of cyclic-element preserving maps vol.160, pp.6, 2013, https://doi.org/10.1016/j.topol.2013.02.002