Bonn Topology Group - Abstracts

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12 December 2017
ARTHUR SOULIÉ (Université de Strasbourg): Long-Moody construction and polynomial functors


In 1994, Long and Moody gave a construction on representations of braid groups which associates a representation of Bn with a representation of Bn+1. This construction complexifies in a sense the initial representation : for instance, starting from a dimension one representation, one obtains the unreduced Burau representation. In this talk, I will present this construction and its generalizations from a functorial point of view. I will explain that each construction of Long and Moody defines an endofunctor, called a Long-Moody functor, on a suitable category of functors. I will give generalizations of this construction to other families of groups such as automorphism groups of free groups, mapping class groups of orientable and non-orientable surfaces or map- ping class groups of 3-manifolds. After defining notions of polynomial functors in this context, I will prove that the Long-Moody functors increase by one the degree of polynomiality. Thus, the Long-Moody constructions will provide new examples of twisted coefficients corresponding to the framework developed by Randal-Williams and Wahl in 2015 to prove homological stability with certain twisted coefficients for different families of groups, in particular the aforementioned ones.

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