Basic constructions in rational homotopy theory of function spaces
[Constructions basiques en théorie d’homotopie rationnelle des espaces fonctionnels]
Buijs, Urtzi1 ; Murillo, Aniceto2
1 Universidad de Málaga Departamento de Algebra Geometría y Topología Ap. 59, 29080 Málaga (Spain)
2 Departamento de Algebra, Geometría y Topología, Universidad de Málaga, Ap. 59, 29080 Málaga, Spain
Annales de l'Institut Fourier, Tome 56 (2006) no. 3, pp. 815-838.

Moyennant le foncteur de réalisation de Bousfield-Gugenheim, et à l’aide comme point de départ du modèle de Brown Szczarba d’un espace de fonctions, on décrit les objets basiques et les applications relatives au type d’homotopie rationnelle des espaces fonctionnels et de leurs composantes arc-connexes.

Via the Bousfield-Gugenheim realization functor, and starting from the Brown-Szczarba model of a function space, we give a functorial framework to describe basic objects and maps concerning the rational homotopy type of function spaces and its path components.

DOI : 10.5802/aif.2201
Classification : 55P62, 54C35
Keywords: Function space, mapping space, Sullivan model, rational homotopy theory
Mot clés : Espace fonctionnel, modèle de Sullivan, homotopie rationnelle
Buijs, Urtzi 1 ; Murillo, Aniceto 2

1 Universidad de Málaga Departamento de Algebra Geometría y Topología Ap. 59, 29080 Málaga (Spain)
2 Departamento de Algebra, Geometría y Topología, Universidad de Málaga, Ap. 59, 29080 Málaga, Spain 
@article{AIF_2006__56_3_815_0,
     author = {Buijs, Urtzi and Murillo, Aniceto},
     title = {Basic constructions in rational homotopy theory of function spaces},
     journal = {Annales de l'Institut Fourier},
     pages = {815--838},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {56},
     number = {3},
     year = {2006},
     doi = {10.5802/aif.2201},
     zbl = {1122.55008},
     mrnumber = {2244231},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.2201/}
}
TY  - JOUR
AU  - Buijs, Urtzi
AU  - Murillo, Aniceto
TI  - Basic constructions in rational homotopy theory of function spaces
JO  - Annales de l'Institut Fourier
PY  - 2006
SP  - 815
EP  - 838
VL  - 56
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://www.numdam.org/articles/10.5802/aif.2201/
DO  - 10.5802/aif.2201
LA  - en
ID  - AIF_2006__56_3_815_0
ER  - 
%0 Journal Article
%A Buijs, Urtzi
%A Murillo, Aniceto
%T Basic constructions in rational homotopy theory of function spaces
%J Annales de l'Institut Fourier
%D 2006
%P 815-838
%V 56
%N 3
%I Association des Annales de l’institut Fourier
%U https://www.numdam.org/articles/10.5802/aif.2201/
%R 10.5802/aif.2201
%G en
%F AIF_2006__56_3_815_0
Buijs, Urtzi; Murillo, Aniceto. Basic constructions in rational homotopy theory of function spaces. Annales de l'Institut Fourier, Tome 56 (2006) no. 3, pp. 815-838. doi : 10.5802/aif.2201. https://www.numdam.org/articles/10.5802/aif.2201/

[1] Bousfield, A. K.; Gugenheim, V. K. A. M. On PL De Rham theory and rational homotopy type, 179 (8), Memoirs of the Amer. Math. Soc., 1976 | MR | Zbl

[2] Brown, E. H.; Szczarba, R. H. Continuous cohomology and real homotopy type, Trans. Amer. Math. Soc., Volume 31 (1989), pp. 57-106 | DOI | MR | Zbl

[3] Brown, E. H.; Szczarba, R. H. On the rational homotopy type of function spaces, Trans. Amer. Math. Soc., Volume 349 (1997), pp. 4931-4951 | DOI | MR | Zbl

[4] Félix, Y. Rational category of the space of sections of a nilpotent bundle, Comment. Math. Helvetici, Volume 65 (1990), pp. 615-622 | DOI | MR | Zbl

[5] Félix, Y.; Halperin, S.; Thomas, J.C. Rational Homotopy Theory, G.T.M., Volume 205, Springer, 2000 | MR | Zbl

[6] Goerss, P. G.; Jardine, J. F. Simplicial Homotopy Theory, Progress in Mathematics, Volume 174, Birkhäuser, Basel-Boston-Berlin, 1999 | MR | Zbl

[7] Haefliger, A. Rational homotopy of the space of sections of a nilpotent bundle, Trans. Amer. Math. Soc., Volume 273 (1982), pp. 609-620 | DOI | MR | Zbl

[8] Halperin, S. Lecture on Minimal Models, 230, Mémoires de la Société Mathématique de France, 1983 | Numdam | Zbl

[9] Hilton, P.; Mislin, G.; Roitberg, J. Localization of nilpotent groups and spaces, 15, North Holland Mathematical Studies, 1975 (North Holland) | MR | Zbl

[10] Kuribayashi, K. A rational model for the evaluation map (2005) (Preprint) | Zbl

[11] May, J. P. Simplicial Objects in Algebraic Topology, Chicago Lectures in Mathematics, 1992 (University of Chicago Press) | MR | Zbl

[12] Milnor, J. On spaces having the homotopy type of a CW-complex, Trans. Amer. Math. Soc., Volume 342 (1994) no. 2, pp. 895-915

[13] Smith, S. Rational homotopy of the space of self-maps of complexes with finitely many homotopy groups, Trans. Amer. Math. Soc., Volume 90 (1994), pp. 272-280 | Zbl

[14] Smith, S. Rational evaluation subgroups, Math. Zeit., Volume 221 (1996), pp. 387-400 | MR | Zbl

[15] Sullivan, D. Infinitesimal computations in Topology, Publ. Math. de l’I.H.E.S., Volume 47 (1978), pp. 269-331 | Numdam | MR | Zbl

[16] Thom, R. L’homologie des espaces fonctionnels, Colloque de topologie algébrique, Centre Belge Rech. Math. (1957), pp. 29-39 | MR | Zbl

[17] Vigué-Poirrier, M. Sur l’homotopie rationnelle des espaces fonctionnels, Manuscripta Math., Volume 56 (1986), pp. 177-191 | DOI | MR | Zbl

Cité par Sources :