Volume 7 (2024)
Palmer, Martin; Soulié, Arthur
Topological representations of motion groups and mapping class groups – a unified functorial construction
Annales Henri Lebesgue, Volume 7 (2024), pp. 409-519.

Metadata

DOI10.5802/ahl.204
Keywords Homological representations, mapping class groups, surface braid groups, loop braid groups, motion groups, Lawrence–Bigelow representations

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.


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