DOI QR코드

DOI QR Code

Evaluation Subgroups of Mapping Spaces over Grassmann Manifolds

  • Abdelhadi Zaim (Department of Mathematics and Computer Sciences, Faculty of Sciences Ain Chock, University Hassan II)
  • Received : 2021.11.13
  • Accepted : 2022.05.09
  • Published : 2023.03.31

Abstract

Let Vk,n (ℂ) denote the complex Steifel and Grk,n (ℂ) the Grassmann manifolds for 1 ≤ k < n. In this paper, we compute, in terms of the Sullivan minimal models, the evaluation subgroups and, more generally, the relative evaluation subgroups of the fibration p : Vk,k+n (ℂ) → Grk,k+n (ℂ). In particular, we prove that G* (Grk,k+n (ℂ), Vk,k+n (ℂ) ; p) is isomorphic to Grel* (Grk,k+n (ℂ), Vk,k+n (ℂ) ; p) ⊕ G* (Vk,k+n (ℂ)).

Keywords

Acknowledgement

The author is very grateful to the referee for carefully reading the manuscript and his helpful suggestions.

References

  1. Y. Felix, S. Halperin and J. -C. Thomas, Rational Homotopy Theory, Graduate Texts in Mathematics, 205, Springer-Verlag, New York(2001). 
  2. Y. Felix, J. Oprea and D. Tanre, Algebraic Models in Geometry 17, Oxford Graduate Texts in Mathematics(2008). 
  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. G. Lupton and S. B. Smith, Rationalized evaluation subgroups of a map I: Sullivan models, derivations and G-sequences, J. Pure Appl. Algebra, 209(1)(2007), 159-171.  https://doi.org/10.1016/j.jpaa.2006.05.018
  5. O. Maphane, Rationalized evaluation subgroups of the complex Hopf fibration, Commun. Korean Math. Soc., 36(4)(2021), 835-840. 
  6. O. Maphane, Derivations of a Sullivan model and the Rationalized G-Sequence, International J. of Math. and Math. Sci., Volume 2021. 
  7. S. B. Smith, Rational evaluation subgroups, Math. Z., 221(3)(1996), 387-400.  https://doi.org/10.1007/PL00004252
  8. D. Sullivan, Infinitesimal computations in topology, Publ. Math. IHES., 47 (1977), 269-331.  https://doi.org/10.1007/BF02684341
  9. M. H. Woo and J. -R. Kim, Certain subgroups of homotopy groups, J. Korean Math. Soc., 21(2)(1984), 109-120. 
  10. M. H. Woo and K. Y. Lee, On the relative evaluation subgroups of a CW-pair, J. Korean Math. Soc., 25(1988), 149-160. 
  11. A. Zaim, Relative Gottlieb groups of mapping spaces and their rational cohomology, Proc. Int. Geom. Cent., 15(1)(2022), 1-15. https://doi.org/10.15673/tmgc.v15i1.2196