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

Palmer, Martin; Soulié, Arthur

doi:10.5802/ahl.204

Topological representations of motion groups and mapping class groups – a unified functorial construction
[Représentations topologiques des groupes de mouvement et des groupes de difféotopie – Une construction fonctorielle unifiée]

Palmer, Martin ¹ ; Soulié, Arthur ²

¹Institutul de Matematică Simion Stoilow al Academiei Române, 21 Calea Griviței, 010702 Bucharest (Romania)
²Normandie Univ., UNICAEN, CNRS, LMNO, 14000 Caen (France)

Annales Henri Lebesgue, Tome 7 (2024), pp. 409-519

Abstract (VO)
Résumé

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.

Une source majeure de représentations intéressantes et grandement non-triviales pour les groupes ayant une origine topologique, tels que les groupes de tresses et les groupes de difféotopie, est donnée par leurs actions sur de l’homologie tordue d’espaces associés ; celles-ci sont connues sous le nom de représentations homologiques. Les représentations de ce type se sont montrées particulièrement importantes pour les questions de linéarité, un exemple clef étant celui de la famille de représentations définies topologiquement par Lawrence et Bigelow, et utilisée par Bigelow et Krammer pour démontrer que les groupes de tresses sont linéaires. Dans cet article, nous établissons des fondations unifiées pour la construction de représentations homologiques en utilisant des méthodes fonctorielles. Plus précisément, nous introduisons des foncteurs de représentations homologiques, qui encodent de larges classes de représentations homologiques, et qui sont définis sur des catégories contenant tous les groupes de difféotopie et tous les groupes de mouvement pour une dimension fixée. Ces catégories sources sont définies à partir d’un enrichissement topologique de la construction de support due à Quillen, que l’on applique à des catégories de variétés décorées. Cette approche unifie de nombreuses constructions déjà connues, y compris celles de Lawrence et Bigelow, et produit beaucoup de nouvelles représentations.

Reçu le : 2021-11-17
Révisé le : 2023-12-01
Accepté le : 2024-01-15
Publié le : 2024-09-05

MR Zbl

DOI : 10.5802/ahl.204

Classification : 20C12, 20F36, 57K20, 18B40, 20C07, 20J05, 55R80, 57M07, 57M10
Keywords: Homological representations, mapping class groups, surface braid groups, loop braid groups, motion groups, Lawrence–Bigelow representations

Affiliations des auteurs :

Palmer, Martin ¹ ; Soulié, Arthur ²

¹ Institutul de Matematică Simion Stoilow al Academiei Române, 21 Calea Griviței, 010702 Bucharest (Romania)
² Normandie Univ., UNICAEN, CNRS, LMNO, 14000 Caen (France)

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Palmer, Martin and Souli\'e, Arthur},
     title = {Topological representations of motion groups and mapping class groups {\textendash} a unified functorial construction},
     journal = {Annales Henri Lebesgue},
     pages = {409--519},
     year = {2024},
     publisher = {\'ENS Rennes},
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     doi = {10.5802/ahl.204},
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Palmer, Martin; Soulié, Arthur. Topological representations of motion groups and mapping class groups – a unified functorial construction. Annales Henri Lebesgue, Tome 7 (2024), pp. 409-519. doi: 10.5802/ahl.204

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