arXiv subjects

Arthur Soulié

Publications and source records attributed to Arthur Soulié.

12 recordsLinked to original sources

arXiv2026

Twisted homological stability for handlebody mapping class groups

We prove twisted homological stability for handlebody mapping class groups. Using the categorical framework developed by Randal-Williams and Wahl, we establish that the homology of the handlebody groups stabilises with respect to both genus and the number of marked boundary discs, for all coefficient systems of finite degree. Our first main theorem refines and extends the twisted stability result for handlebodies outlined by Randal-Williams and Wahl, allowing any number of marked discs and boundary points. We then introduce the notion of coefficient bisystem to treat stability under variation of boundary markings. As an application, we deduce homological stability for moduli spaces of 3-dimensional handlebodies equipped with tangential structures.

math.GT↗

arXiv2025

The pro-nilpotent Lawrence-Krammer-Bigelow representation

We construct a 3-variable enrichment of the Lawrence-Krammer-Bigelow (LKB) representation of the braid groups, which is the limit of a pro-nilpotent tower of representations having the original LKB representation as its bottom layer. We also construct analogous pro-nilpotent towers of representations of surface braid groups and loop braid groups.

math.GT↗

arXiv2024

Polynomiality of surface braid and mapping class group representations

We study a wide range of homologically-defined representations of surface braid groups and of mapping class groups of surfaces, including the Lawrence-Bigelow representations of the classical braid groups. These representations naturally come in families, defining homological representation functors on categories associated to surface braid groups or all mapping class groups. We prove that many of these homological representation functors are polynomial. This has applications to twisted homological stability and to understanding the structure of the representation theory of the associated families of groups. Our polynomiality results are consequences of more fundamental results establishing relations amongst the coherent representations that we consider via short exact sequences of functors. As well as polynomiality, these short exact sequences also have applications to understanding the kernels of the homological representations under consideration.

math.GT↗

arXiv2024

Stable twisted cohomology of the mapping class groups in the unit tangent bundle homology

We compute the stable cohomology groups of the mapping class groups of compact orientable surfaces with one boundary, with twisted coefficients given by the homology of the unit tangent bundle of the surface. This stable twisted cohomology is not free as a module over the stable cohomology algebra with constant coefficients. In fact, it is out of the scope of the traditional framework for twisted cohomological stability, since these twisted coefficients define a covariant functor over the classical category associated to mapping class groups to study homological stability, rather than a contravariant one. For comparison, we also compute the stable cohomology group with coefficients in the first cohomology of the unit tangent bundle of the surface, which fits into the traditional framework.

math.GR↗

arXiv2024

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

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.

math.AT↗

arXiv2023

Stable twisted cohomology of the mapping class groups in the exterior powers of the unit tangent bundle homology

We study the stable cohomology groups of the mapping class groups of surfaces with twisted coefficients given by the $d^{th}$ exterior powers of the first rational homology of the unit tangent bundles of the surfaces $\tilde{H}_{\mathbb{Q}}$. These coefficients are outside of the traditional framework of cohomological stability. They form a module $H_{\mathrm{st}}^{*}(Λ^{d}\tilde{H}_{\mathbb{Q}})$ over the stable cohomology algebra of the mapping class groups with trivial coefficients denoted by $\mathrm{Sym}_{\mathbb{Q}}(\mathcal{E})$. If $d\neq 2$, the $\mathrm{Tor}$-group in each degree of $H_{\mathrm{st}}^{*}(Λ^{d}\tilde{H}_{\mathbb{Q}})$ does not vanish, and we compute all these $\mathrm{Tor}$-groups explicitly for $d \leq 5$. In particular, for each $d\neq 2$, the module $H_{\mathrm{st}}^{*}(Λ^{d}\tilde{H}_{\mathbb{Q}})$ is not free over $\mathrm{Sym}_{\mathbb{Q}}(\mathcal{E})$, while it is free for $d=2$.For comparison, we also compute the stable cohomology group with coefficients in the $d^{th}$ exterior powers of the first rational cohomology of the unit tangent bundle of the surface, which fit into the classical framework of cohomological stability.

math.AC↗

arXiv2023

Extensions of Tong-Yang-Ma representation

In 1996, Tong, Yang and Ma defined a family of representations of the braid group which have the same dimensions as the (unreduced) Burau representations but are not equivalent. The Burau representation was defined homologically and extended to the string links in several ways. In this paper, using the method of Silver and Williams, we extend the family of the Tong-Yang-Ma representations to the string links and welded string links. Moreover, we show that the kernels of these representations may be described using some linking numbers. Finally, we apply the Long-Moody construction to the Tong-Yang-Ma representations and study its first properties.

math.GT↗

arXiv2022

When the lower central series stops: a comprehensive study for braid groups and their relatives

Understanding the lower central series of a group is, in general, a difficult task. It is, however, a rewarding one: computing the lower central series and the associated Lie algebras of a group or of some of its subgroups can lead to a deep understanding of the underlying structure of that group. Our goal here is to showcase several techniques aimed at carrying out part of this task. In particular, we seek to answer the following question: when does the lower central series stop? We introduce a number of tools that we then apply to various groups related to braid groups: the braid groups themselves, surface braid groups, groups of virtual and welded braids, and partitioned versions of all of these groups. The path from our general techniques to their application is far from being a straight one, and some astuteness and tenacity is required to deal with all of the cases encountered along the way. Nevertheless, we arrive at an answer to our question for each and every one of these groups, save for one family of partitioned braid groups on the projective plane. In several cases, we even compute completely the lower central series. Some results about the lower central series of Artin groups are also included.

math.GT↗

arXiv2022

The Burau representations of loop braid groups

We give a simple topological construction of the Burau representations of the loop braid groups. There are four versions: defined either on the non-extended or extended loop braid groups, and in each case there is an unreduced and a reduced version. Three are not surprising, and one could easily guess the correct matrices to assign to generators. The fourth is more subtle, and does not seem combinatorially obvious, although it is topologically very natural.

math.GT↗

arXiv2021

Some computations of stable twisted homology for mapping class groups

In this paper, we deal with stable homology computations with twisted coefficients for mapping class groups of surfaces and of 3-manifolds, automorphism groups of free groups with boundaries and automorphism groups of certain right-angled Artin groups. On the one hand, the computations are led using semidirect product structures arising naturally from these groups. On the other hand, we compute the stable homology with twisted coefficients by FI-modules. This notably uses a decomposition result of the stable homology with twisted coefficients for pre-braided monoidal categories proved in this paper.

math.AT↗

arXiv2021

Generalized Long-Moody functors

In this paper, we generalize the principle of the Long-Moody construction for representations of braid groups to other groups, such as mapping class groups of surfaces. Namely, we introduce endofunctors over a functor category that encodes representations of a family of groups. They are called Long-Moody functors and provide new representations. In this context, notions of polynomial functors are defined and play an important role in the study of homological stability. We prove that, under additional assumptions, a Long-Moody functor increases the very strong and weak polynomial degrees of functors by one.

math.AT↗

arXiv2020

A note on representations of welded braid groups

In this note, we adapt the procedure of the Long-Moody procedure to construct linear representations of welded braid groups. We exhibit the natural setting in this context and compute the first examples of representations we obtain thanks to this method. We take this way also the opportunity to review the few known linear representations of welded braid groups.

math.GR↗