Martin David Palmer -- Preprint -- A unified functorial construction of homological representations of families of groups
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A unified functorial construction of homological representations of families of groups
with Arthur Soulié
(v1: 2019)

arXiv: abspdf
v1

Abstract

The families of braid groups, surface braid groups, mapping class groups and loop braid groups have a representation theory of "wild type", so it is very useful to be able to construct such representations topologically, so that they may be understood by topological or geometric methods. For the braid groups Bn, Lawrence and Bigelow have constructed families of representations starting from actions of Bn on the twisted homology of configuration spaces. These were then used by Bigelow and Krammer to prove that the braid groups are linear.

We develop a general underlying procedure to build homological representations of families of groups, encompassing all of the above-mentioned families and in principle many more, such as families of general motion groups. Moreover, these families of representations are coherent, in the sense that they extend to a functor on a larger category, whose automorphism groups are the family of groups under consideration and whose richer structure may be used (i) to organise the representation theory of the family of groups and (ii) to prove twisted homological stability results &emdash; both via the notion of polynomiality. We prove polynomiality for many such homological functors, including those (which we construct) extending the Lawrence-Bigelow representations.

This helps to unify previously-known constructions and to produce new families of representations &emdash; we do this for the loop braid groups, surface braid groups and mapping class groups. In particular, for the loop braid groups, we construct three analogues of the Lawrence-Bigelow representations (of the classical braid groups), which appear to be new.