Long-Moody functors and stable homology of mapping class groups

Abstract : Among the linear representations of braid groups, Burau representations are recovered from a trivial representation using a construction introduced by Long in 1994, following a collaboration with Moody. This construction, called the Long-Moody construction, thus allows to construct more and more complex representations of braid groups. In this thesis, we have a functorial point of view on this construction, which allows find more easily some variants. Moreover, the degree of polynomiality of a functor measures its complexity. We thus show that the Long-Moody construction defines a functor LM, which increases the degree of polynomiality. Furthermore, we define analogous functors for other families of groups such as mapping class groups of surfaces and 3-manifolds, symmetric groups or automorphism groups of free groups. They satisfy similar properties on the polynomiality. Hence, Long-Moody functors provide twisted coefficients fitting into the framework of the homological stability results of Randal-Williams and Wahl for the afore mentioned families of groups. Finally, we give a comparison result for 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 deals with their effect on the polynomiality. The first one deals with the generalisation of Long-Moody functors for other families of groups. The last chapter touches on stable homology computations for mapping class group.
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Arthur Soulié. Long-Moody functors and stable homology of mapping class groups. Algebraic Topology [math.AT]. Université de Strasbourg, 2018. English. ⟨NNT : 2018STRAD016⟩. ⟨tel-01819086v4⟩

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