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On the rational homotopy type of function spaces. (English) Zbl 0927.55012

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 Sullivan’s minimal models for \(X\) and \(Y\). Such constructions appeared in papers of A. Haefliger (1982), M. Vigué-Poirrier (1986), A. Bousfield, C. Peterson and L. Smith (1989), all of them in the bibliography. A version mixing coalgebras and Lie algebras is also performed in [H. Scheerer and D. Tanré, Arch. Math. 59, No. 2, 130-145 (1992; Zbl 0729.55008].

MSC:

55P15 Classification of homotopy type
55P62 Rational homotopy theory
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[2] A. K. Bousfield, C. Peterson, and L. Smith, The rational homology of function spaces, Arch. Math. (Basel) 52 (1989), no. 3, 275 – 283. · Zbl 0674.55008
[3] Edgar H. Brown Jr. and Robert H. Szczarba, Continuous cohomology and real homotopy type, Trans. Amer. Math. Soc. 311 (1989), no. 1, 57 – 106. · Zbl 0671.55006
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[8] Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269 – 331 (1978). · Zbl 0374.57002
[9] Micheline Vigué-Poirrier and Dennis Sullivan, The homology theory of the closed geodesic problem, J. Differential Geometry 11 (1976), no. 4, 633 – 644. · Zbl 0361.53058
[10] Micheline Vigué-Poirrier, Cohomologie de l’espace des sections d’un fibré et cohomologie de Gelfand-Fuchs d’une variété, Algebra, algebraic topology and their interactions (Stockholm, 1983) Lecture Notes in Math., vol. 1183, Springer, Berlin, 1986, pp. 371 – 396 (French). · Zbl 0591.55004
[11] Micheline Vigué-Poirrier, Sur l’homotopie rationnelle des espaces fonctionnels, Manuscripta Math. 56 (1986), no. 2, 177 – 191 (French, with English summary). · Zbl 0597.55008
[12] G.W. Whitehead, On products in homotopy groups, Ann. Math. 47 (1946), 460-475. · Zbl 0060.41106
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