Opérades cellulaires et espaces de lacets itérés
Annales de l'Institut Fourier, Tome 46 (1996) no. 4, pp. 1125-1157.

L’espace des configurations de p points distincts de R admet une filtration naturelle qui est induite par les inclusions des R n dans R . Nous caractérisons le type d’homotopie de cette filtration par les propriétés combinatoires d’une structure cellulaire sous-jacente, étroitement liée à la théorie des E n -opérades de May. Cela donne une approche unifiée des différents modèles combinatoires d’espaces de lacets itérés et redémontre les théorèmes d’approximation de Milgram, Smith et Kashiwabara.

The configuration space of p-tuples of pairwise distinct points in R carries a natural filtration coming from the inclusions of the R n into R . We characterize the homotopy type of this filtration by the combinatorial properties of an underlying cellular structure and establish a close relationship to May’s theory of E n -operads. This gives a unified approach to the different known combinatorial models of iterated loop spaces reproving by the way the approximation theorems of Milgram, Smith and Kashiwabara.

@article{AIF_1996__46_4_1125_0,
     author = {Berger, Clemens},
     title = {Op\'erades cellulaires et espaces de lacets it\'er\'es},
     journal = {Annales de l'Institut Fourier},
     pages = {1125--1157},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {46},
     number = {4},
     year = {1996},
     doi = {10.5802/aif.1543},
     mrnumber = {1415960},
     zbl = {0853.55007},
     language = {fr},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1543/}
}
TY  - JOUR
AU  - Berger, Clemens
TI  - Opérades cellulaires et espaces de lacets itérés
JO  - Annales de l'Institut Fourier
PY  - 1996
SP  - 1125
EP  - 1157
VL  - 46
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1543/
DO  - 10.5802/aif.1543
LA  - fr
ID  - AIF_1996__46_4_1125_0
ER  - 
%0 Journal Article
%A Berger, Clemens
%T Opérades cellulaires et espaces de lacets itérés
%J Annales de l'Institut Fourier
%D 1996
%P 1125-1157
%V 46
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1543/
%R 10.5802/aif.1543
%G fr
%F AIF_1996__46_4_1125_0
Berger, Clemens. Opérades cellulaires et espaces de lacets itérés. Annales de l'Institut Fourier, Tome 46 (1996) no. 4, pp. 1125-1157. doi : 10.5802/aif.1543. https://aif.centre-mersenne.org/articles/10.5802/aif.1543/

[1] C. Balteanu, Z. Fiedorowicz, R. Schwänzl et R. Vogt, Iterated monoidal categories, preprint (1995).

[2] H.-J. Baues, Geometry of loop spaces and the cobar-construction, Mem. of the Amer. Math. Soc., 230 (1980). | MR | Zbl

[3] M. G. Barratt et P. J. Eccles, Γ+-Structures I, II, III, Topology, 13 (1974), 25-45, 113-126, 199-207. | Zbl

[4] C. Berger, Un groupoïde simplicial comme modèle de l'espace des chemins, Bull. Soc. Math. France, 123 (1995), 1-32. | Numdam | MR | Zbl

[5] R. Blind et P. Mani, On puzzles and polytope isomorphism, Aequationes Math., 34 (1987), 287-297. | MR | Zbl

[6] J. M. Boardman et R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Math. 347, Springer Verlag (1973). | MR | Zbl

[7] F.R. Cohen, The homology of Cn+1-spaces, Lecture Notes in Math. 533, Springer Verlag (1976), 207-351. | MR | Zbl

[8] F.R. Cohen, J. P. May et L. R. Taylor, Splitting of certain spaces CX, Math. Proc. Camb. Phil. Soc., 84 (1978), 465-496. | MR | Zbl

[9] G. Dunn, Tensor product of operads and iterated loop spaces, Journal of Pure and Applied Algebra, 50 (1988), 237-258. | MR | Zbl

[10] R. Fox et L. Neuwirth, The braid groups, Math. Scand., 10 (1962), 119-126. | MR | Zbl

[11] M.M. Kapranov, The permutoassociahedron, MacLane's coherence theorem and asymptotic zones for KZ equation, Journal of Pure and Applied Algebra, 85 (1993), 119-142. | Zbl

[12] T. Kashiwabara, On the Homotopy Type of Configuration Complexes, Contemporary Math., 146 (1993), 159-170. | MR | Zbl

[13] J.P. May, The Geometry of iterated loop spaces, Lecture Notes in Math., 277, Springer Verlag (1972). | MR | Zbl

[14] J.P. May, E∞-spaces, group completions and permutative categories, London Math. Soc. Lecture Notes, 11 (1974), 61-94. | Zbl

[15] R. J. Milgram, Iterated loop spaces, Annals of Math., 84 (1966), 386-403. | MR | Zbl

[16] D. Quillen, Higher algebraic K-theory I, Lecture Notes in Math., 341, Springer Verlag (1973), 85-147. | MR | Zbl

[17] G. Segal, Configuration spaces and iterated loop spaces, Inventiones Math., 21 (1973), 213-221. | MR | Zbl

[18] J.H. Smith, Simplicial Group Models for ΩnSnX, Israel Journal of Math., 66 (1989), 330-350. | MR | Zbl

Cité par Sources :