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

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.

Final version, to appear in the Annales Henri Lebesgue. 110 pages. Significantly revised and reorganised from v4

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• Representations of braid groups
• Homological stability of topological moduli spaces
• Problems, Questions, and Conjectures about Mapping Class Groups
• Homological stability for moduli spaces of disconnected submanifolds, I
• Mapping class groups are not linear in positive characteristic
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Cited by in corpus (6)

• The Burau representations of loop braid groups
• Lawrence-Bigelow representations, bases and duality
• When the lower central series stops: a comprehensive study for braid groups and their relatives
• Polynomiality of surface braid and mapping class group representations
• On the homology of big mapping class groups
• The pro-nilpotent Lawrence-Krammer-Bigelow representation