Journal of the Mathematical Society of Japan

L models of based mapping spaces

Urtzi BUIJS, Yves FÉLIX, and Aniceto MURILLO

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Abstract

In this paper, for any pointed map f: XY between finite type nilpotent CW-complexes, we obtain L and Lie models of mapf*(X,Y), the pointed space of based maps homotopic to f, in terms of Lie algebras constructed from the Quillen models of X and Y. The main advantage of our approach is to allow X to be an infinite dimensional CW-complex, in which case mapf*(X,Y) has no longer the homotopy type of a finite type CW-complex.

Article information

Source
J. Math. Soc. Japan, Volume 63, Number 2 (2011), 503-524.

Dates
First available in Project Euclid: 25 April 2011

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1303737796

Digital Object Identifier
doi:10.2969/jmsj/06320503

Mathematical Reviews number (MathSciNet)
MR2793109

Zentralblatt MATH identifier
1252.55006

Subjects
Primary: 55P62: Rational homotopy theory 54C35: Function spaces [See also 46Exx, 58D15]

Keywords
mapping space L_∞ algebra Quillen model rational homotopy theory

Citation

BUIJS, Urtzi; FÉLIX, Yves; MURILLO, Aniceto. L ∞ models of based mapping spaces. J. Math. Soc. Japan 63 (2011), no. 2, 503--524. doi:10.2969/jmsj/06320503. https://projecteuclid.org/euclid.jmsj/1303737796


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References

  • J. Block and A. Lazarev, André-Quillen cohomology and rational homotopy of function spaces, Adv. in Math., 193 (2005), 18–39.
  • A. K. Bousfield and V. K. A. M. Gugenheim, On PL De Rahm theory and rational homotopy type, Mem. Amer. Math. Soc., 179 (1976).
  • A. K. Bousfiled and D. M. Kan, Homotopy Limits, Completions and Localizations, Springer LNM 304, 1972.
  • E. H. Brown and R. H. Szczarba, On the rational homotopy type of function spaces, Trans. Amer. Math. Soc., 349 (1997), 4931–4951.
  • U. Buijs and A. Murillo, The rational homotopy Lie algebra of function spaces, Comment. Math. Helv., 83 (2008), 723–739.
  • U. Buijs, Y. Félix and A. Murillo, Lie models for the components of sections of a nilpotent fibration, to appear in Trans. Amer. Math. Soc.
  • U. Buijs, Y. Félix and A. Murillo, Rational homotopy of the (homotopy) fixed point sets of circle actions, Adv. in Math., 222 (2009), 151–171.
  • M. Chas and D. Sullivan, String topology, to appear in Ann. of Math.
  • Y. Félix, S. Halperin and J. C. Thomas, Rational homotopy theory, Springer GTM 205, 2000.
  • E. Getzler, Lie theory for nilpotent $L_\infty$-algebras, Ann. of Math., 170 (2009), 271–301.
  • A. Haefliger, Rational homotopy of the space of sections of a nilpotent bundle, Trans. Amer. Math. Soc., 273 (1982), 609–620.
  • A. Henriques, Integrating $L_\infty$-algebras, Compositio Mathematica, 144 (2008), 1017–1045.
  • T. V. Kadeishvili, Algebraic structure in the homology of an $A(\infty)$-algebra, (Russian. ENglish summary), Soobshch. Akad. Nauk. Gruz. SSR, 108 (1982), 249–252.
  • M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys., 66 (2003), 157–216.
  • K. Kuribayashi and T. Yamaguchi, A rational splitting of a based mapping space, Alg. and Geom. Topology, 6 (2006), 309–327.
  • T. Lada and M. Markl, Strongly homotopy Lie algebras, Comm. in Algebra, 23 (1995), 2147–2161.
  • T. Lada and J. Stasheff, Introduction to sh algebras for physicists, Int. J. Theor. Phys., 32 (1993), 1087–1104.
  • G. Lupton and S. B. Smith, Whitehead products in function spaces: Quillen model formulae, J. Math. Soc. Japan, 62 (2010), 49–81.
  • M. Majewski, Rational homotopical models and uniqueness, Mem. Amer. Math. Soc., 682 (2000).
  • J. M. Möller, Nilpotent spaces of sections, Trans. Amer. Math. Soc., 303 (1987), 733–741.
  • D. Quillen, Rational homotopy theory, Ann. of Math., 90 (1969), 205–295.
  • M. Shlessinger and J. Stasheff, The Lie algebra structure of tangent cohomology and deformation theory, Jour. Pure Appl. Algebra, 38 (1985), 313–322.
  • D. Sullivan, Infinitesimal computations in topology, Publ. Math. de l'I.H.E.S., 47 (1978), 269–331.
  • D. Tanré, Homotopie rationnelle: modèles de Chen, Quillen, Sullivan, Springer LNM 1025, 1983.
  • R. Thom, L'homologie des espaces fonctionnels, Colloque de topologie algébrique, Louvain, (1957), 29–39.
  • C. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, Cambridge, 1994.