Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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RATIONAL HOMOTOPY TYPE OF MAPPING SPACES BETWEEN COMPLEX PROJECTIVE SPACES AND THEIR EVALUATION SUBGROUPS

  • Gatsinzi, Jean-Baptiste (Department of Mathematics and Statistical Sciences Botswana International University of Science and Technology)
  • Received : 2020.11.16
  • Accepted : 2021.01.20
  • Published : 2022.01.31
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Abstract

We use L∞ models to compute the rational homotopy type of the mapping space of the component of the natural inclusion in,k : ℂPn ↪ ℂPn+k between complex projective spaces and show that it has the rational homotopy type of a product of odd dimensional spheres and a complex projective space. We also characterize the mapping aut1 ℂPn → map(ℂPn, ℂPn+k; in,k) and the resulting G-sequence.

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References

  1. J. Block and A. Lazarev, Andre-Quillen cohomology and rational homotopy of function spaces, Adv. Math. 193 (2005), no. 1, 18-39. https://doi.org/10.1016/j.aim.2004.04.014
  2. E. H. Brown, Jr., and R. H. Szczarba, On the rational homotopy type of function spaces, Trans. Amer. Math. Soc. 349 (1997), no. 12, 4931-4951. https://doi.org/10.1090/S0002-9947-97-01871-0
  3. U. Buijs, Y. Felix, and A. Murillo, Lie models for the components of sections of a nilpotent fibration, Trans. Amer. Math. Soc. 361 (2009), no. 10, 5601-5614. https://doi.org/10.1090/S0002-9947-09-04870-3
  4. U. Buijs, Y. Felix, and A. Murillo, L∞ models of based mapping spaces, J. Math. Soc. Japan 63 (2011), no. 2, 503-524. http://projecteuclid.org/euclid.jmsj/1303737796 https://doi.org/10.2969/jmsj/06320503
  5. U. Buijs, Y. Felix, and A. Murillo, L∞ rational homotopy of mapping spaces, Rev. Mat. Complut. 26 (2013), no. 2, 573-588. https://doi.org/10.1007/s13163-012-0105-z
  6. Y. Felix, S. Halperin, and J.-C. Thomas, Rational homotopy theory, Graduate Texts in Mathematics, 205, Springer-Verlag, New York, 2001. https://doi.org/10.1007/978-1-4613-0105-9
  7. J.-B. Gatsinzi, On the genus of elliptic fibrations, Proc. Amer. Math. Soc. 132 (2004), no. 2, 597-606. https://doi.org/10.1090/S0002-9939-03-07203-4
  8. J.-B. Gatsinzi and R. Kwashira, Rational homotopy groups of function spaces, in Homotopy theory of function spaces and related topics, 105-114, Contemp. Math., 519, Amer. Math. Soc., Providence, RI, 2010. https://doi.org/10.1090/conm/519/10235
  9. D. H. Gottlieb, Evaluation subgroups of homotopy groups, Amer. J. Math. 91 (1969), 729-756. https://doi.org/10.2307/2373349
  10. A. Haefliger, Rational homotopy of the space of sections of a nilpotent bundle, Trans. Amer. Math. Soc. 273 (1982), no. 2, 609-620. https://doi.org/10.2307/1999931
  11. T. Lada and M. Markl, Strongly homotopy Lie algebras, Comm. Algebra 23 (1995), no. 6, 2147-2161. https://doi.org/10.1080/00927879508825335
  12. G. Lupton and S. B. Smith, Rationalized evaluation subgroups of a map. I. Sullivan models, derivations and G-sequences, J. Pure Appl. Algebra 209 (2007), no. 1, 159-171. https://doi.org/10.1016/j.jpaa.2006.05.018
  13. G. Lupton and S. B. Smith, Rationalized evaluation subgroups of a map. II. Quillen models and adjoint maps, J. Pure Appl. Algebra 209 (2007), no. 1, 173-188. https://doi.org/10.1016/j.jpaa.2006.05.019
  14. S. Mac Lane, Homology, Die Grundlehren der mathematischen Wissenschaften, Bd. 114, Academic Press, Inc., Publishers, New York, 1963.
  15. J. M. Moller and M. Raussen, Rational homotopy of spaces of maps into spheres and complex projective spaces, Trans. Amer. Math. Soc. 292 (1985), no. 2, 721-732. https://doi.org/10.2307/2000242
  16. D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Etudes Sci. Publ. Math. No. 47 (1977), 269-331 (1978).
  17. 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.