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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the rational homotopy type
of function spaces


Authors: Edgar H. Brown Jr. and Robert H. Szczarba Jr.
Journal: Trans. Amer. Math. Soc. 349 (1997), 4931-4951
MSC (1991): Primary 55P15, 55P62
DOI: https://doi.org/10.1090/S0002-9947-97-01871-0
MathSciNet review: 1407482
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Abstract: The main result of this paper is the construction of a minimal model for the function space $\mathcal {F}(X,Y)$ of continuous functions from a finite type, finite dimensional space $X$ to a finite type, nilpotent space $Y$ in terms of minimal models for $X$ and $Y$. For the component containing the constant map, $\pi _{*}(\mathcal {F}(X,Y))\otimes Q =\pi _{*}(Y)\otimes H^{-*}(X;Q)$ in positive dimensions. When $X$ is formal, there is a simple formula for the differential of the minimal model in terms of the differential of the minimal model for $Y$ and the coproduct of $H_{*}(X;Q)$. We also give a version of the main result for the space of cross sections of a fibration.


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  • [BG] A.K. Bousfield and V.K.A.M. Gugenheim, On $PL$ de Rham theory and rational homotopy type, Mem. Amer. Math. Soc. No. 179 (1976). MR 54:13906
  • [BPS] A. Bousfield, C. Peterson, and L. Smith, The rational homology of function spaces, Arch. Math 52 (1989), 275-283. MR 90d:55020
  • [BZ] E.H. Brown and R.H. Szczarba, Continuous cohomology and real homotopy type, Trans. AMS 311 (1989), 57-106. MR 89f:55005
  • [H] A. Haefliger, Rational homotopy of the space of sections of a nilpotent bundle, Trans. AMS 273 (1982), 609-620. MR 84a:55010
  • [Hi] P. Hilton, G. Mislin, and J. Roitberg, Localization of nilpotent groups and spaces, North Holland Mathematics Studies 15, North Holland, New York, 1975. MR 57:17635
  • [L] J. Lannes, Sur la cohomologie modulo $p$ des $p$-groupes abéliens élémentaires, Proc. Durham Symp. on Homotopy Theory 1985, LMS Lecture Note Ser., no. 117, Camb. Univ. Press, 1987, 97-116. MR 89e:55037
  • [M] J.P. May, Simplicial objects in algebraic topology, Univ. Chicago Press, Chicago IL., 1992. MR 93m:55025
  • [S] D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Etudes Sci. Publ. Math. 47 (1977), 269-331. MR 58:31119
  • [VS] M. Vigué-Poirrier and D. Sullivan, The homology theory of the closed geodesic problem, J. Differential Geometry 11 (1976), 633-644. MR 56:13269
  • [V1] M. Vigué-Poirrier, Cohomologie de l'espace des sections d'un fibre et cohomologie de Gelfand-Fuchs d'une variete, Lecture Notes in Math., 1183, Springer-Verlag, 1986, 371-396. MR 87i:55026
  • [V2] -, Sur l'homotopie rationnelle des espaces functionnels, Manuscripta Math. 56 (1986), 177-191. MR 87h:55009
  • [W] G.W. Whitehead, On products in homotopy groups, Ann. Math. 47 (1946), 460-475. MR 8:50b

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Additional Information

Edgar H. Brown Jr.
Affiliation: Department of Mathematics, Brandeis University, Waltham, Massachusetts 02254

Robert H. Szczarba Jr.
Affiliation: Department of Mathematics, Yale University, Box 208283, New Haven, Connecticut 06520

DOI: https://doi.org/10.1090/S0002-9947-97-01871-0
Received by editor(s): February 12, 1996
Article copyright: © Copyright 1997 American Mathematical Society

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