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

• Published in: Annales Henri Lebesgue Vol. 7; pp. 409 - 519
• Main Authors: Palmer, Martin, Soulié, Arthur
• Format: Journal Article
• Language: English
• Published: UFR de Mathématiques - IRMAR 05.09.2024
• Subjects: Algebraic Topology; Category Theory; Geometric Topology; Group Theory; Mathematics; Representation Theory; 2020 Mathematics Subject Classification: Primary: 20C12; 20J05; 57K20; Secondary: 18B40; loop braid groups; surface braid groups; mapping class groups; 55R80; Homological representations; Lawrence-Bigelow representations; 57M07; 57M10 Homological representations; Lawrence–Bigelow representations; 20C07; 20F36; motion groups; 2020 Mathematics Subject Classification: Primary: 20C12 20F36 57K20; Secondary: 18B40 20C07 20J05 55R80 57M07 57M10 Homological representations mapping class groups surface braid groups loop braid groups motion groups Lawrence-Bigelow representations
• ISSN: 2644-9463; 2644-9463
• DOI: 10.5802/ahl.204

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.