Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Martin Palmer, Arthur Soulié

TL;DR

The paper develops a unified, functorial framework for constructing homological representations of groups of topological origin, notably braid and mapping class groups, by viewing these groups in families and modelling their actions on twisted homology of associated embedding spaces. It builds a topology-enhanced categorical foundation using decorated manifolds and a topologically enriched Quillen bracket construction, enabling coherent, potentially twisted representations across families via two main semifunctors and a universal coefficient system. The approach recovers classical Lawrence–Bigelow representations for braids and Moriyama/An–Ko-type representations in other settings, while also generating numerous new representations with twisted coefficients and novel ground rings, and it sets the stage for polynomiality and stability analyses in a subsequent work. By applying the construction to motion groups, surface braid groups, loop braid groups, and mapping class groups, the work provides a flexible, global framework that connects geometry, topology, and linear representations, offering a structured path toward understanding linearity and stability phenomena in a broad class of groups.

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.

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

TL;DR

The paper develops a unified, functorial framework for constructing homological representations of groups of topological origin, notably braid and mapping class groups, by viewing these groups in families and modelling their actions on twisted homology of associated embedding spaces. It builds a topology-enhanced categorical foundation using decorated manifolds and a topologically enriched Quillen bracket construction, enabling coherent, potentially twisted representations across families via two main semifunctors and a universal coefficient system. The approach recovers classical Lawrence–Bigelow representations for braids and Moriyama/An–Ko-type representations in other settings, while also generating numerous new representations with twisted coefficients and novel ground rings, and it sets the stage for polynomiality and stability analyses in a subsequent work. By applying the construction to motion groups, surface braid groups, loop braid groups, and mapping class groups, the work provides a flexible, global framework that connects geometry, topology, and linear representations, offering a structured path toward understanding linearity and stability phenomena in a broad class of groups.

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.

Paper Structure

This paper contains 44 sections, 6 theorems, 146 equations, 4 figures, 1 table.

Table of Contents

  1. Categories of decorated manifolds and their embeddings
  2. Decorated manifolds and their diffeomorphisms
  3. Split short exact sequences
  4. The first short exact sequence
  5. The second short exact sequence
  6. Connecting the short exact sequences
  7. Motion groups and braided mapping class groups
  8. Actions of motion groups
  9. A Serre fibration of decorated diffeomorphism groups
  10. The topologically-enriched Quillen bracket construction
  11. A topological enrichment of the Quillen bracket construction
  12. Quillen bracket categories of manifolds
  13. Functorial split short exact sequences
  14. Skeleta of Quillen bracket categories
  15. Construction of homological representation functors
  16. ...and 29 more sections

Key Result

Theorem A

For a fixed set of parameters $\{i,Z,\mathsf{G},\ell\}$ as described above, the action of the family of groups $\{g_{n} \colon G_{n} \to G_{n+1}\}_{n\in\mathbb{N}}$ on the twisted homology of certain embedding spaces determines functors where $\mathcal{Q}$ denotes a group built out of the deck transformation groups of the regular covering spaces corresponding to the coefficients in the twisted ho

Figures (4)

  • Figure 1: An illustration of the notation for the solid cylinder $\mathbb{B}^d_t$ from Notation \ref{['notation:cylinder']} for $d=3$. Its lower boundary $\partial_{l} \mathbb{B}^3_t$ is coloured yellow, its base $b\mathbb{B}^3_t$ is yellow-green and the codimension-$2$ stratum $\partial (b\mathbb{B}^3_t) = \partial \mathbb{D}^{2} \times \{0\}$ is blue.
  • Figure 2: Two decorated manifolds and their boundary connected sum.
  • Figure 3: The boundary connected sum $L \natural M$ from the proof of Theorem \ref{['thm:fibre-bundle']}.
  • Figure 4: The submanifold $M_{\epsilon,t}$ (shaded in green) of $L \natural M$ from the proof of Theorem \ref{['thm:fibre-bundle']}.

Theorems & Definitions (117)

  • Theorem A: Theorems \ref{['thm:construction']}, \ref{['thm:global_functor_motion_groups']} and \ref{['thm:global_functor_mcg']}
  • Theorem B: Corollary \ref{['cor:description_UD_d']}
  • Theorem C: Theorem \ref{['thm:LawrenceBigelowFunctors']}, §\ref{['ss:applications_motion_groups']}
  • Theorem D: Example \ref{['eg:An-Ko']}, §\ref{['ss:applications_motion_groups']}
  • Theorem E: Example \ref{['eg:loop_Burau']}, §\ref{['ss:applications_motion_groups']}
  • Theorem F: Proposition \ref{['prop:Moriyama_recovering']}, §\ref{['ss:mcg_construction']}
  • Remark 1
  • Remark 4
  • proof
  • Remark 5
  • ...and 107 more