Topological representations of motion groups and mapping class groups -- a unified functorial construction

TL;DR

This paper develops a unified, functorial framework for constructing homological representations of topological groups like braid and mapping class groups, encompassing many known and new representations.

Contribution

It introduces a functorial approach to generate a broad class of homological representations, unifying previous constructions and enabling new representations.

Findings

Provides a unified foundation for homological representations

Includes many known constructions as special cases

Generates new representations for mapping class and motion groups

Abstract

For groups of a topological origin, such as braid groups and mapping class groups, an important source of interesting and highly non-trivial representations is given by their actions on the twisted homology of associated spaces; these are known as homological representations. Representations of this kind have proved themselves especially important for the question of linearity, a key example being the family of topologically-defined representations introduced by Lawrence and Bigelow, and used by Bigelow and Krammer to prove that braid groups are linear. In this paper, we give a unified foundation for the construction of homological representations using a functorial approach. Namely, we introduce homological representation functors encoding a large class of homological representations, defined on categories containing all mapping class groups and motion groups in a fixed dimension. These source categories are defined using a topological enrichment of the Quillen bracket construction applied to categories of decorated manifolds. This approach unifies many previously-known constructions, including those of Lawrence-Bigelow, and yields many new representations.