Opérades cellulaires et espaces de lacets itérés

Berger, Clemens

Annales de l'Institut Fourier, Tome 46 (1996) no. 4, pp. 1125-1157.

L’espace des configurations de $p$ points distincts de ${\mathbf{R}}^{\infty }$ admet une filtration naturelle qui est induite par les inclusions des ${\mathbf{R}}^n$ dans ${\mathbf{R}}^{\infty }$. 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 ${\mathbf{R}}^{\infty }$ carries a natural filtration coming from the inclusions of the ${\mathbf{R}}^n$ into ${\mathbf{R}}^{\infty }$. 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.

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