Martin David Palmer -- Preprint -- A unified functorial construction of homological representations of families of groups
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Topological representations of motion groups and mapping class groups -- a unified functorial construction
with Arthur Soulié
(v1: 2019, v3: 2020)

arXiv: abspdf
v3v2v1

Abstract

Braid groups, mapping class groups and similar groups of geometric origin typically have "wild" representation theory. One would therefore like to construct representations of these groups by topological or geometric means, in order to be able to understand them with topological or geometric tools. As one very important example, Lawrence and Bigelow have constructed families of linear representations of the classical braid groups starting from actions on the twisted homology of configuration spaces, which were then used by Bigelow and Krammer to prove the linearity of the braid groups.

We give a unified construction of such topological representations: in each dimension d, we construct a large family of representations of a category UDd whose automorphism groups contain all mapping class groups and motion groups in dimension d. There are three parameters that one may vary in the construction: a submanifold Z ⊂ Rd and two integers ℓ ≥ 2 and i ≥ 0. In particular, varying the parameter ℓ leads to a "pro-nilpotent" tower of representations, of which we give several non-trivial examples in dimensions 2 and 3. The richer structure of the category UDd (beyond its automorphisms) may moreover be used to organise the representation theory of the family of groups in question.

This recovers and unifies many previously-known constructions, including those of Lawrence-Bigelow, as well as the Long-Moody construction, using an iterative variant of our construction. We also discuss some of the new families of representations that we obtain in dimensions 2 and 3, for the surface braid groups, loop braid groups and mapping class groups.