no. 4
Opérades cellulaires et espaces de lacets itérés
Annales de l'Institut Fourier, Volume 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.

Berger, Clemens. Opérades cellulaires et espaces de lacets itérés. Annales de l'Institut Fourier, Volume 46 (1996) no. 4, pp. 1125-1157. doi: 10.5802/aif.1543
@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},
     year = {1996},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {46},
     number = {4},
     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

[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). | Zbl | MR

[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. | Zbl | MR | Numdam

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

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

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

[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. | Zbl | MR

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

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

[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. | Zbl | MR

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

[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. | Zbl | MR

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

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

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

Cited by Sources: