Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

GENERALIZED GOTTLIEB SUBGROUPS AND SERRE FIBRATIONS

  • Published : 2009.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

Let ${\pi}:E{\rightarrow}B$ be a Serre fibration with fibre F. We prove that if the inclusion map $i:F{\rightarrow}E$ has a left homotopy inverse r and ${\pi}:E{\rightarrow}B$ admits a cross section ${\rho}:B{\rightarrow}E$, then $G_n(E,F){\cong}{\pi}_n(B){\oplus}G_n(F)$. This is a generalization of the case of trivial fibration which has been proved by Lee and Woo in [8]. Using this result, we will prove that ${\pi}_n(X^A){\cong}{\pi}_n(X){\oplus}G_n(F)$ for the function space $X^A$ from a space A to a weak $H_*$-space X where the evaluation map ${\omega}:X^A{\rightarrow}X$ is regarded as a fibration.

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. B. Gray, Homotopy Theory, Academic Press, New York, 1975.
  4. Y. Hirato, K. Kuribayashi, and N. Oda, A function space model approach to the rational evaluation subgroups, Math. Z. 258 (2008), no. 3, 521-555. https://doi.org/10.1007/s00209-007-0184-6
  5. J. R. Kim, Localizations and generalized evaluation subgroups of homotopy groups, J. Korean Math. Soc. 22 (1985), no. 1, 9-18.
  6. J. R. Kim and M. H. Woo, Certain subgroups of homotopy groups, J. Korean Math. Soc. 21 (1984), no. 2, 109-120.
  7. S. S. Koh, Note on the homotopy properties of the components of the mapping space $X^{s^p}$, Proc. Amer. Math. Soc. 11 (1960), 896-904.
  8. K. Y. Lee and M. H. Woo, Generalized evaluation subgroups of product spaces relative to a factor, Proc. Amer. Math. Soc. 124 (1996), no. 7, 2255-2260. https://doi.org/10.1090/S0002-9939-96-03588-5
  9. 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
  10. K. Y. Lee and M. H. Woo, On the relative evaluation subgroups of a CW-pair, J. Korean Math. Soc. 25 (1988), no. 1, 149-160.
  11. K. L. Lim, On cyclic maps, J. Austral. Math. Soc. Ser. A 32 (1982), no. 3, 349-357. https://doi.org/10.1017/S1446788700024903
  12. K. Varadarajian, Generalised Gottlieb groups, J. Indian Math. Soc. (N.S.) 33 (1969), 141-164.
  13. H. Wada, Note on some mapping spaces, Tohoku Math. J. (2) 10 (1958), 143-145.