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Theses

Quasimorphismes sur les groupes de tresses et forme de Blanchfield

Abstract : In 2004, Gambaudo and Ghys proved a formula establishing a connection between the ω-signatures of a link and the symplectic features of a representation of the braid group. Their main motivation was the construction on quasimorphisms on homeomorphism and diffeomorphism groups.The main goal of this thesis is to extend this result in terms of an algebraic invariant of a braids: the Witt class of the Blanchfield form. Some link invariants are defined through the cyclic covering spaces of their exterior. (Co)homology groups are then equipped with module structures over the ring Λ = Z[π]. For example, the Blanchfield form of a link is a generalisation of the linking form of a 3-manifold, which is a bilinear form on the torsionpart of its first homology group. In particular, every braid β defines a class L(β) in a Witt group WT(Λ) .Theorem. Let α and β be two braids. Then, in WT(Λ):L(αβ) - L(α) - L(β) = -∂ Meyer(Burau(α), Burau(β)),where the Meyer cocycle is defined on the sub group of GLn(Λ) whose elements preserve the Squier form.The result by Gambaudo and Ghys can essentially be recovered from this equality.
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Maxime Bourrigan. Quasimorphismes sur les groupes de tresses et forme de Blanchfield. Mathématiques générales [math.GM]. Ecole normale supérieure de lyon - ENS LYON, 2013. Français. ⟨NNT : 2013ENSL0831⟩. ⟨tel-00872081⟩

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