## 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

### Keywords:

minimal model; function space

### Citations:

Zbl 0760.55009; Zbl 0729.55008
Full Text:

### References:

 [1] A. K. Bousfield and V. K. A. M. Gugenheim, On \?\? de Rham theory and rational homotopy type, Mem. Amer. Math. Soc. 8 (1976), no. 179, ix+94. · Zbl 0338.55008 [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 [4] André Haefliger, Rational homotopy of the space of sections of a nilpotent bundle, Trans. Amer. Math. Soc. 273 (1982), no. 2, 609 – 620. · Zbl 0508.55019 [5] Peter Hilton, Guido Mislin, and Joe Roitberg, Localization of nilpotent groups and spaces, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. North-Holland Mathematics Studies, No. 15; Notas de Matemática, No. 55. [Notes on Mathematics, No. 55]. · Zbl 0323.55016 [6] J. Lannes, Sur la cohomologie modulo \? des \?-groupes abéliens élémentaires, Homotopy theory (Durham, 1985) London Math. Soc. Lecture Note Ser., vol. 117, Cambridge Univ. Press, Cambridge, 1987, pp. 97 – 116 (French). [7] J. Peter May, Simplicial objects in algebraic topology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992. Reprint of the 1967 original. · Zbl 0769.55001 [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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