×

\(L_{\infty }\) models of based mapping spaces. (English) Zbl 1252.55006

Higher homotopy Lie algebras, known as strongly homotopy Lie algebras, abbreviated sh-Lie algebras and denoted by \(L_{\infty}\)-algebras, have been introduced as a generalization of Lie algebras and Lie superalgebras. \(L_{\infty}\) algebras were introduced as a model for Lie algebras that satisfy the Jacobi identity up to all higher homotopies and they reflect more accurately the homotopy type of a space.
Models of a mapping space \(\text{map}^*_f(X,Y)\) have been obtained in different contexts by various authors when \(X\) is a finite type complex.
In the present paper, the authors study the Lie model and \(L_{\infty}\) model of the mapping space \(\text{map}^*_f(X,Y)\) which in this case is no longer of finite type. They explicitly describe and give the Lie model and the \(L_{\infty}\) model of \(\text{map}^*_f(X,Y)\) as the suspension of Lie algebras of derivations.

MSC:

55P62 Rational homotopy theory
54C35 Function spaces in general topology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. Block and A. Lazarev, André-Quillen cohomology and rational homotopy of function spaces, Adv. in Math., 193 (2005), 18-39. · Zbl 1070.55006
[2] A. K. Bousfield and V. K. A. M. Gugenheim, On PL De Rahm theory and rational homotopy type, Mem. Amer. Math. Soc., 179 (1976). · Zbl 0338.55008
[3] A. K. Bousfiled and D. M. Kan, Homotopy Limits, Completions and Localizations, Springer LNM 304 , 1972. · Zbl 0259.55004
[4] E. H. Brown and R. H. Szczarba, On the rational homotopy type of function spaces, Trans. Amer. Math. Soc., 349 (1997), 4931-4951. · Zbl 0927.55012
[5] U. Buijs and A. Murillo, The rational homotopy Lie algebra of function spaces, Comment. Math. Helv., 83 (2008), 723-739. · Zbl 1169.55006
[6] 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. · Zbl 1180.55008
[7] 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. · Zbl 1175.55010
[8] M. Chas and D. Sullivan, String topology, to appear in Ann. of Math. · Zbl 1068.55009
[9] Y. Félix, S. Halperin and J. C. Thomas, Rational homotopy theory, Springer GTM 205 , 2000. · Zbl 1325.55001
[10] E. Getzler, Lie theory for nilpotent \(L_\infty\)-algebras, Ann. of Math., 170 (2009), 271-301. · Zbl 1246.17025
[11] A. Haefliger, Rational homotopy of the space of sections of a nilpotent bundle, Trans. Amer. Math. Soc., 273 (1982), 609-620. · Zbl 0508.55019
[12] A. Henriques, Integrating \(L_\infty\)-algebras, Compositio Mathematica, 144 (2008), 1017-1045. · Zbl 1152.17010
[13] 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. · Zbl 0535.55005
[14] M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys., 66 (2003), 157-216. · Zbl 1058.53065
[15] K. Kuribayashi and T. Yamaguchi, A rational splitting of a based mapping space, Alg. and Geom. Topology, 6 (2006), 309-327. · Zbl 1097.55010
[16] T. Lada and M. Markl, Strongly homotopy Lie algebras, Comm. in Algebra, 23 (1995), 2147-2161. · Zbl 0999.17019
[17] T. Lada and J. Stasheff, Introduction to sh algebras for physicists, Int. J. Theor. Phys., 32 (1993), 1087-1104. · Zbl 0824.17024
[18] G. Lupton and S. B. Smith, Whitehead products in function spaces: Quillen model formulae, J. Math. Soc. Japan, 62 (2010), 49-81. · Zbl 1193.55005
[19] M. Majewski, Rational homotopical models and uniqueness, Mem. Amer. Math. Soc., 682 (2000). · Zbl 0942.55015
[20] J. M. Möller, Nilpotent spaces of sections, Trans. Amer. Math. Soc., 303 (1987), 733-741. · Zbl 0628.55007
[21] D. Quillen, Rational homotopy theory, Ann. of Math., 90 (1969), 205-295. · Zbl 0191.53702
[22] M. Shlessinger and J. Stasheff, The Lie algebra structure of tangent cohomology and deformation theory, Jour. Pure Appl. Algebra, 38 (1985), 313-322. · Zbl 0576.17008
[23] D. Sullivan, Infinitesimal computations in topology, Publ. Math. de l’I.H.E.S., 47 (1978), 269-331. · Zbl 0374.57002
[24] D. Tanré, Homotopie rationnelle: modèles de Chen, Quillen, Sullivan, Springer LNM 1025 , 1983. · Zbl 0539.55001
[25] R. Thom, L’homologie des espaces fonctionnels, Colloque de topologie algébrique, Louvain, (1957), 29-39. · Zbl 0077.36301
[26] C. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38 , Cambridge University Press, Cambridge, 1994. · Zbl 0797.18001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.