Work

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

Year: 2024

Type: article

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.

Source: Annales Henri Lebesgue

DOI: 10.5802/ahl.204

Authors: Martin Palmer, Arthur Soulié

Institutions: IMAR - Institutul de Matematică „Simion Stoilow” al Academiei Române, Laboratoire de Mathématiques Nicolas Oresme

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funders: University of Glasgow, Agence Nationale de la Recherche, Unitatea Executiva pentru Finantarea Invatamantului Superior, a Cercetarii, Dezvoltarii si Inovarii, Institute for Basic Science