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Etude de l’opérade Swiss-cheese et applications à la théorie des longs noeuds

Abstract : The aim of this work is to study the Swiss-Cheese operad, denoted by SCd, which is a relative version of the little cubes operad Cd.We show that the classical theorems in the context of uncolored operads can begeneralized to the relative case. From a pointed operad O (i.e. a two colored operad under π0(SC₁) ), webuild two semi-cosimplicial spaces (Oc ; Oo) such that the pair of semi-totalizations is weakly equivalentto an explicit SC₂-algebra. In particular, we prove that the pair (ℒ₁ ; n ; ℒm; n), composed of the space oflong knots and the space of long links, is weakly equivalent to an explicit SC₂-algebra.We study two homology theories, namely singular and Hochschild homology, of a pair of semicosimplicialspaces arising from a pointed operad. In this context, (H∗(sTot(Oc)) ; H∗(sTot(Oo))) and (HH∗(Oc) ; HH∗(Oo)) are equipped with an explicit H∗(SC₂)-algebra structure. We show that the mapintroduced by Bousfield between these two pairs is a morphism of H∗(SC₂)-algebras. This result helps us to understand the pair of spectral sequences computing (H∗(sTot(Oc)) ; H∗(sTot(Oo))). In particular wegive some conditions on a multiplicative symmetric operad so that the E² pages of the Bousfield spectral sequences are weakly equivalent to H∗(sTot(Oc)) and H∗(sTot(Oo)) as H∗(SC₂)-algebras. Finally we generalize our previous results, relying on a conjecture by Dwyer and Hess. We define acolored operad CCd and obtain an SCd₊₁-algebra from an operad morphism CCd → O. As a consequence, we prove that the couple of topological spaces (ℒᵈ₁ ; n ; T∞Imm(ᴷ))(Rᵈ ; Rⁿ)), where Ld₁;n is the space of long knots from Rd to Rⁿ and where T∞Imm(k)(Rᵈ ; Rⁿ) is the polynomial approximation of the (k)-immersions,is weakly equivalent to an explicit SCd+₁-algebra.
Keywords : Model Category
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Julien Ducoulombier. Etude de l’opérade Swiss-cheese et applications à la théorie des longs noeuds. Topologie générale [math.GN]. Université Sorbonne Paris Cité, 2015. Français. ⟨NNT : 2015USPCD090⟩. ⟨tel-02128412⟩

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