SMF - Publications - Séminaires et Congrès - Titles

Operads 2009
VERSION IMPRIMEE A VENIR - PRINT VERSION NOT YET AVAILABLE
Loday, Jean-Louis and Vallette, Bruno, eds
Séminaires et Congrès 26 (2013), 281 pages
Presentation, Summary

The (non-equivariant) homology of the little disks operad
Dev P. Sinha
Séminaires et Congrès 26 (2013), 253-279

Résumé :
L'homologie (non équivariante) de l'opérade des petits disques
Dans cet article d'exposition, nous calculons explicitement et de manière élémentaire l'homologie de l'opérade des petits disques. Ceci implique que l'homologie d'un espace de lacets fois itérés est une algèbre de Poisson. En démontrant ce résultat, nous cherchons à familiariser le lecteur avec les espaces de configurations de points dans un espace euclidien et ce en utilisant des outils élémentaires. Les nouveaux résultats portent sur l'identification de la dualité entre l'homologie et la cohomologie de ces espaces à l'aide d'une dualité entre certains graphes et arbres ainsi qu'en utilisant la structure de coopérade des groupes de cohomologie.

Abstract:
In this expository paper we give an elementary, explicit computation of the homology of the little disks operad, which then shows that the homology of a -fold loop space is a Poisson algebra. In the course of proving this basic result, we aim to familiarize a larger audience with Euclidean configuration spaces, using elementary tools. New results include identifying the pairing between homology and cohomology of these spaces as a pairing of graphs and trees, and treating the cooperad structure on cohomology.

Keywords: Little disks operads, configurations spaces, Poisson (co)operad.

Class. math. : 18D50, 55P48, 55R80.

ISBN : 978-2-85629-363-8
ISSN : 1285-2783

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