Formalité opéradique et homotopie des espaces de configuration

Idrissi, Najib

Type de document :

Thèse

Titre :

Formalité opéradique et homotopie des espaces de configuration

Titre en anglais :

Operadic formality and homotopy of configuration spaces

Auteur(s) :

Idrissi, Najib [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]

Directeur(s) de thèse :

Benoit Fresse

Date de soutenance :

2017-11-17

Président du jury :

Damien Calaque
Kathryn Hess
Pascal Lambrechts
Patrick Popescu-Pampu
Thomas Willwacher

Membre(s) du jury :

Damien Calaque
Kathryn Hess
Pascal Lambrechts
Patrick Popescu-Pampu
Thomas Willwacher

Organisme de délivrance :

Université Lille 1

École doctorale :

Sciences pour l'ingénieur (ED 072)

Mot(s)-clé(s) :

opérades
espaces de configuration (topologie)
variétés topologiques
topologie algébrique

Mot(s)-clé(s) en anglais :

operads
configuration spaces (topology)
topological manifolds
algebraic topology

Discipline(s) HAL :

Mathématiques [math]/Topologie algébrique [math.AT]

Résumé en anglais : [en]

In a first part, we study Voronov’s "Swiss-Cheese" operad SC2, which governs the action of a D2-algebra on a D1-algebra. We build a model in groupoids of this operad and we describe algebras over this model in a manner ...
Lire la suite >In a first part, we study Voronov’s "Swiss-Cheese" operad SC2, which governs the action of a D2-algebra on a D1-algebra. We build a model in groupoids of this operad and we describe algebras over this model in a manner similar to the classical description of algebras over H*(SC). We extend our model into a rational model which depends on a Drinfeld associator, and we compare this new model to the one that we would get if the operad SC were formal. In a second part, we study configuration spaces of closed smooth simply connected manifolds. We prove over R a conjecture of Lambrechts–Stanley which describes a mode of such configuration spaces, and we obtain as corollary their real homotopy invariance. Moreover, using Kontsevich’s proof of the formality of the operads Dn, we obtain that this model is compatible with the action of the Fulton–MacPherson operad when the manifold is framed. This allows us to explicitly compute the factorization homology of such a manifold. Finally, in a third part, we expand this result to a large class of manifolds with boundary. We first use a chain-level Poincaré–Lefschetz duality result to compute the homology of the configuration spaces of these manifolds, then we reuse the methods of the second chapter to obtain our model, which is compatible with the action of the Swiss-Cheese operad SCn.Lire moins >

Langue :

Français

Collections :

Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL

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