Long-Moody functors and stable homology of mapping class groups.

Abstract : The Burau representations of braid groups can be obtained from on a trivial representation applying a construction due to Long in 1994, as a result of a collaboration with Moody. This construction, called the Long-Moody construction, allows to build increasingly complex representations of braid groups. In this thesis, we have a functorial point of on the Long-Moody construction, which allows to introduce some variants. Moreover, the polynomial degree of a functor measures its complexity. We thus show that the Long-Moody construction defines a functor LM, which increases the very strong polynomial degree. Furthermore, we define analogous functors for other families of groups, such as mapping class groups of surfaces and 3-manifolds, symmetric groups and automorphism groups of free groups.They satisfy similar properties on the polynomiality of a functor. The Long-Moody functors thus provide twisted coefficients conforming with the framework of the homological stability results of Randal-Williams and Wahl for the aforementioned families of groups. Finally, we give a comparison result between the stable homology with coefficient given by a functor F and the one with coefficient given by the functor LM(F) obtained applying a Long-Moody functor. This thesis has three chapters. The first one introduces Long-Moody functors for braid groups and their effect on the polynomiality. The second chapter deals with the generalization of the Long-Moody functors for other families of groups. The second chapter is about stable homology computations for mapping class groups.
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Arthur Soulié. Long-Moody functors and stable homology of mapping class groups.. Algebraic Topology [math.AT]. Institut de Recherche Mathématique Avancée, UMR 7501, 2018. English. ⟨tel-01819086v2⟩

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