arXiv subjects

Gwenael Massuyeau

Publications and source records attributed to Gwenael Massuyeau.

At least 19 recordsLinked to original sources

arXiv2026

On the Sp-structure of the torsion of the Lie algebra of homology cylinders

Let $Σ$ be a compact oriented surface. We investigate the graded Lie algebra of homology cylinders over $Σ$, as introduced by M. Goussarov and K. Habiro. To analyze its torsion, we refine a strategy initiated by Y. Nozaki, M. Suzuki, and the third author, which relies on the reduction modulo $1$ of the LMO functor and on clasper calculus. Specifically, we develop general tools for studying, under the standard action of the symplectic group, the Sp-module structure of the torsion in the odd-degree component of this Lie algebra. We show that this torsion part surjects onto an $2$-torsion Sp-module, which is explicitly described in terms of Jacobi diagrams. As an application, we provide an intrinsic description of the Sp-module given by the degree-three component of the Lie algebra of homology cylinders. A further motivation is to understand torsion phenomena in the associated graded of the lower central series of the Torelli group of $Σ$: in this direction, we exhibit an explicit Sp-module onto which the torsion of the Torelli Lie algebra surjects in degree three.

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arXiv2025

The Johnson-Morita theory for the handlebody group

The Johnson-Morita theory is an algebraic approach to the mapping class group of a surface, in which one considers its action on the successive nilpotent quotients of the fundamental group of the surface. In this paper, we develop an analogue of this theory for the handlebody group, i.e. the mapping class group of a 3-dimensional handlebody. Thus, we obtain a filtration on the handlebody group, prove that its associated graded embeds into a Lie algebra of "special derivations", and give an explicit diagrammatic description of this graded Lie algebra in terms of "oriented trees with beads". Our new diagrammatic method reveals part of the richness of the algebraic structure of the handlebody group, which lies mainly in the subgroup generated by Dehn twists along meridians: the so-called "twist group". As an application, we obtain that each term of the associated graded of the lower central series of the twist group is infinitely generated.

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arXiv2025

On the degree-two part of the associated graded of the lower central series of the Torelli group

We consider the associated graded $\bigoplus_{k\geq 1} Γ_k \mathcal{I} / Γ_{k+1} \mathcal{I} $ of the lower central series $\mathcal{I} = Γ_1 \mathcal{I} \supset Γ_2 \mathcal{I} \supset Γ_3 \mathcal{I} \supset \cdots$ of the Torelli group $\mathcal{I}$ of a compact oriented surface. Its degree-one part is well-understood by D. Johnson's seminal works on the abelianization of the Torelli group. The knowledge of the degree-two part $(Γ_2 \mathcal{I} / Γ_3 \mathcal{I})\otimes \mathbb{Q}$ with rational coefficients arises from works of S. Morita on the Casson invariant and R. Hain on the Malcev completion of $\mathcal{I}$. Here, we prove that the abelian group $Γ_2 \mathcal{I} / Γ_3 \mathcal{I}$ is torsion-free, and we describe it as a lattice in a rational vector space. As an application, the group $\mathcal{I}/Γ_3 \mathcal{I}$ is computed, and it is shown to embed in the group of homology cylinders modulo the surgery relation of $Y_3$-equivalence.

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arXiv2024

On the non-triviality of the torsion subgroup of the abelianized Johnson kernel

The Johnson kernel is the subgroup of the mapping class group of a closed oriented surface that is generated by Dehn twists along separating simple closed curves. The rational abelianization of the Johnson kernel has been computed by Dimca, Hain and Papadima, and a more explicit form was subsequently provided by Morita, Sakasai and Suzuki. Based on these results, Nozaki, Sato and Suzuki used the theory of finite-type invariants of 3-manifolds to prove that the torsion subgroup of the abelianized Johnson kernel is non-trivial. In this paper, we give a purely 2-dimensional proof of the non-triviality of this torsion subgroup and provide a lower bound for its cardinality. Our main tool is the action of the mapping class group on the Malcev Lie algebra of the fundamental group of the surface. Using the same infinitesimal techniques, we also provide an alternative diagrammatic description of the rational abelianized Johnson kernel, and we include in the results the case of an oriented surface with one boundary component.

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arXiv2023

Surgery equivalence relations for 3-manifolds

By classical results of Rochlin, Thom, Wallace and Lickorish, it is well-known that any two 3-manifolds (with diffeomorphic boundaries) are related one to the other by surgery operations. Yet, by restricting the type of the surgeries, one can define several families of non-trivial equivalence relations on the sets of (diffeomorphism classes of) 3-manifolds. In this expository paper, which is based on lectures given at the school ``Winter Braids XI'' (Dijon, December 2021), we explain how certain filtrations of mapping class groups of surfaces enter into the definitions and the mutual comparison of these surgery equivalence relations. We also survey the ways in which concrete invariants of 3-manifolds (such as finite-type invariants) can be used to characterize such relations.

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arXiv2020

The Kontsevich integral for bottom tangles in handlebodies

Using an extension of the Kontsevich integral to tangles in handlebodies similar to a construction given by Andersen, Mattes and Reshetikhin, we construct a functor $Z:\mathcal{B}\to \widehat{\mathbb{A}}$, where $\mathcal{B}$ is the category of bottom tangles in handlebodies and $\widehat{\mathbb{A}}$ is the degree-completion of the category $\mathbb{A}$ of Jacobi diagrams in handlebodies. As a symmetric monoidal linear category, $\mathbb{A}$ is the linear PROP governing "Casimir Hopf algebras", which are cocommutative Hopf algebras equipped with a primitive invariant symmetric 2-tensor. The functor $Z$ induces a canonical isomorphism $\hbox{gr}\mathcal{B} \cong \mathbb{A}$, where $\hbox{gr}\mathcal{B}$ is the associated graded of the Vassiliev-Goussarov filtration on $\mathcal{B}$. To each Drinfeld associator $φ$ we associate a ribbon quasi-Hopf algebra $H_φ$ in $\hbox{gr}\mathcal{B}$, and we prove that the braided Hopf algebra resulting from $H_φ$ by "transmutation" is precisely the image by $Z$ of a canonical Hopf algebra in the braided category $\mathcal{B}$. Finally, we explain how $Z$ refines the LMO functor, which is a TQFT-like functor extending the Le-Murakami-Ohtsuki invariant.

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arXiv2020

A splicing formula for the LMO invariant

We prove a "splicing formula" for the LMO invariant, which is the universal finite-type invariant of rational homology $3$-spheres. Specifically, if a rational homology $3$-sphere $M$ is obtained by gluing the exteriors of two framed knots $K_1 \subset M_1$ and $K_2\subset M_2$ in rational homology $3$-spheres, our formula expresses the LMO invariant of $M$ in terms of the Kontsevich-LMO invariants of $(M_1,K_1)$ and $(M_2,K_2)$. The proof uses the techniques that Bar-Natan and Lawrence developed to obtain a rational surgery formula for the LMO invariant. In low degrees, we recover Fujita's formula for the Casson-Walker invariant and we observe that the second term of the Ohtsuki series is not additive under "standard" splicing. The splicing formula also works when each $M_i$ comes with a link $L_i$ in addition to the knot $K_i$, hence we get a "satellite formula" for the Kontsevich-LMO invariant.

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arXiv2020

Generalized Dehn twists on surfaces and homology cylinders

Let $Σ$ be a compact oriented surface. The Dehn twist along every simple closed curve $γ\subset Σ$ induces an automorphism of the fundamental group $π$ of $Σ$. There are two possible ways to generalize such automorphisms if the curve $γ$ is allowed to have self-intersections. One way is to consider the `generalized Dehn twist' along $γ$: an automorphism of the Malcev completion of $π$ whose definition involves intersection operations and only depends on the homotopy class $[γ]\in π$ of $γ$. Another way is to choose in the usual cylinder $U:=Σ\times [-1,+1]$ a knot $L$ projecting onto $γ$, to perform a surgery along $L$ so as to get a homology cylinder $U_L$, and let $U_L$ act on every nilpotent quotient $π/Γ_{j} π$ of $π$ (where $Γ_jπ$ denotes the subgroup of $π$ generated by commutators of length $j$). In this paper, assuming that $[γ]$ is in $Γ_k π$ for some $k\geq 2$, we prove that (whatever the choice of $L$ is) the automorphism of $π/Γ_{2k+1} π$ induced by $U_L$ agrees with the generalized Dehn twist along $γ$ and we explicitly compute this automorphism in terms of $[γ]$ modulo ${Γ_{k+2}}π$. As applications, we obtain new formulas for certain evaluations of the Johnson homomorphisms showing, in particular, how to realize any element of their targets by some explicit homology cylinders and/or generalized Dehn twists.

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arXiv2019

Generalized Dehn twists in low-dimensional topology

The generalized Dehn twist along a closed curve in an oriented surface is an algebraic construction which involves intersections of loops in the surface. It is defined as an automorphism of the Malcev completion of the fundamental group of the surface. As the name suggests, for the case where the curve has no self-intersection, it is induced from the usual Dehn twist along the curve. In this expository article, after explaining their definition, we review several results about generalized Dehn twists such as their realizability as diffeomorphisms of the surface, their diagrammatic description in terms of decorated trees and the Hopf-algebraic framework underlying their construction. Going to the dimension three, we also overview the relation between generalized Dehn twists and $3$-dimensional homology cobordisms, and we survey the variants of generalized Dehn twists for skein algebras of the surface.

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arXiv2018

Morita's trace maps on the group of homology cobordisms

Morita introduced in 2008 a 1-cocycle on the group of homology cobordisms of surfaces with values in an infinite-dimensional vector space. His 1-cocycle contains all the "traces" of Johnson homomorphisms which he introduced fifteen years earlier in his study of the mapping class group. In this paper, we propose a new version of Morita's 1-cocycle based on a simple and explicit construction. Our 1-cocycle is proved to satisfy several fundamental properties, including a connection with the Magnus representation and the LMO homomorphism. As an application, we show that the rational abelianization of the group of homology cobordisms is non-trivial. Besides, we apply some of our algebraic methods to compare two natural filtrations on the automorphism group of a finitely-generated free group.

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arXiv2018

Brackets in representation algebras of Hopf algebras

For any graded bialgebras $A$ and $B$, we define a commutative graded algebra $A_B$ representing the functor of $B$-representations of $A$. When $A$ is a cocommutative graded Hopf algebra and $B$ is a commutative ungraded Hopf algebra, we introduce a method deriving a Gerstenhaber bracket in $A_B$ from a Fox pairing in $A$ and a balanced biderivation in $B$. Our construction is inspired by Van den Bergh's non-commutative Poisson geometry, and may be viewed as an algebraic generalization of the Atiyah--Bott--Goldman Poisson structures on moduli spaces of representations of surface groups.

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arXiv2017

Brackets in the Pontryagin algebras of manifolds

Given a smooth oriented manifold $M$ with non-empty boundary, we study the Pontryagin algebra $A=H_\ast(Ω)$ where $ Ω$ is the space of loops in $M$ based at a distinguished point of $ \partial M$. Using the ideas of string topology of Chas-Sullivan, we define a linear map $\{\{-,-\}\}: A \otimes A \to A\otimes A$ which is a double bracket in the sense of Van den Bergh satisfying a version of the Jacobi identity. For $\dim(M)\geq 3$, the double bracket $\{\{-,-\}\}$ induces Gerstenhaber brackets in the representation algebras associated with $A$. This extends our previous work on the case $\dim(M)=2$ where $A= H_0(Ω)$ is the group algebra of the fundamental group $π_1(M)$ and the double bracket $\{\{-,-\}\}$ induces the standard Poisson brackets on the moduli spaces of representations of $π_1(M)$.

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arXiv2017

Generalized Johnson homomorphisms for extended N-series

The Johnson filtration of the mapping class group of a compact, oriented surface is the descending series consisting of the kernels of the actions on the nilpotent quotients of the fundamental group of the surface. Each term of the Johnson filtration admits a Johnson homomorphism, whose kernel is the next term in the filtration. In this paper, we consider a general situation where a group acts on a group with a filtration called an "extended N-series". We develop a theory of Johnson homomorphisms in this general setting, including many known variants of the original Johnson homomorphisms as well as several new variants.

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arXiv2017

A functorial extension of the Magnus representation to the category of three-dimensional cobordisms

Let $R$ be an integral domain and $G$ be a subgroup of its group of units. We consider the category $\mathbf{\mathsf{Cob}}_G$ of 3-dimensional cobordisms between oriented surfaces with connected boundary, equipped with a representation of their fundamental group in $G$. Under some mild conditions on $R$, we construct a monoidal functor from $\mathbf{\mathsf{Cob}}_G$ to the category $\mathbf{\mathsf{pLagr}}_R$ consisting of "pointed Lagrangian relations" between skew-Hermitian $R$-modules. We call it the "Magnus functor" since it contains the Magnus representation of mapping class groups as a special case. Our construction is inspired from the work of Cimasoni and Turaev on the extension of the Burau representation of braid groups to the category of tangles. It can also be regarded as a $G$-equivariant version of a TQFT-like functor that has been described by Donaldson. The study and computation of the Magnus functor is carried out using classical techniques of low-dimensional topology. When $G$ is a free abelian group and $R = Z[G]$ is the group ring of $G$, we relate the Magnus functor to the "Alexander functor" (which has been introduced in a prior work using Alexander-type invariants), and we deduce a factorization formula for the latter.

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arXiv2016

Formal descriptions of Turaev's loop operations

Using intersection and self-intersection of loops, Turaev introduced in the seventies two fundamental operations on the algebra $\mathbb{Q}[π]$ of the fundamental group $π$ of a surface with boundary. The first operation is binary and measures the intersection of two oriented based curves on the surface, while the second operation is unary and computes the self-intersection of an oriented based curve. It is already known that Turaev's intersection pairing has an algebraic description when the group algebra $\mathbb{Q}[π]$ is completed with respect to powers of its augmentation ideal and is appropriately identified to the degree-completion of the tensor algebra $T(H)$ of $H:=H_1(π;\mathbb{Q})$. In this paper, we obtain a similar algebraic description for Turaev's self-intersection map in the case of a disk with $p$ punctures. Here we consider the identification between the completions of $\mathbb{Q}[π]$ and $T(H)$ that arises from a Drinfeld associator by embedding $π$ into the pure braid group on $(p+1)$ strands; our algebraic description involves a formal power series which is explicitly determined by the associator. The proof is based on some three-dimensional formulas for Turaev's loop operations, which involve $2$-strand pure braids and are shown for any surface with boundary.

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arXiv2014

A functorial extension of the abelian Reidemeister torsions of three-manifolds

Let $\mathbb{F}$ be a field and let $G\subset \mathbb{F}\setminus \{0\}$ be a multiplicative subgroup. We consider the category $\mathcal{Cob}_G$ of $3$-dimensional cobordisms equipped with a representation of their fundamental group in $G$, and the category ${Vect}_{\mathbb{F},\pm G}$ of $\mathbb{F}$-linear maps defined up to multiplication by an element of $\pm G$. Using the elementary theory of Reidemeister torsions, we construct a "Reidemeister functor" from $Cob_G$ to ${Vect}_{\mathbb{F},\pm G}$. In particular, when the group $G$ is free abelian and $\mathbb{F}$ is the field of fractions of the group ring $\mathbb{Z}[G]$, we obtain a functorial formulation of an Alexander-type invariant introduced by Lescop for $3$-manifolds with boundary; when $G$ is trivial, the Reidemeister functor specializes to the TQFT developed by Frohman and Nicas to enclose the Alexander polynomial of knots. The study of the Reidemeister functor is carried out for any multiplicative subgroup $G\subset \mathbb{F}\setminus \{0\}$. We obtain a duality result and we show that the resulting projective representation of the monoid of homology cobordisms is equivalent to the Magnus representation combined with the relative Reidemeister torsion.

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arXiv2014

Splitting formulas for the LMO invariant of rational homology three-spheres

For rational homology 3-spheres, there exist two universal finite-type invariants: the Le-Murakami-Ohtsuki invariant and the Kontsevich-Kuperberg-Thurston invariant. These invariants take values in the same space of "Jacobi diagrams", but it is not known whether they are equal. In 2004, Lescop proved that the KKT invariant satisfies some "splitting formulas" which relate the variations of KKT under replacement of embedded rational homology handlebodies by others in a "Lagrangian-preserving" way. We show that the LMO invariant satisfies exactly the same relations. The proof is based on the LMO functor, which is a generalization of the LMO invariant to the category of 3-dimensional cobordisms, and we generalize Lescop's splitting formulas to this setting.

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arXiv2013

Lefschetz fibrations, intersection numbers, and representations of the framed braid group

We examine the action of the fundamental group $Γ$ of a Riemann surface with $m$ punctures on the middle dimensional homology of a regular fiber in a Lefschetz fibration, and describe to what extent this action can be recovered from the intersection numbers of vanishing cycles. Basis changes for the vanishing cycles result in a nonlinear action of the framed braid group $\widetilde{\mathcal B}$ on $m$ strings on a suitable space of $m\times m$ matrices. This action is determined by a family of cohomologous 1-cocycles ${\mathcal S}_c:\widetilde{\mathcal B}\to GL_m({\mathbb{Z}}[Γ])$ parametrized by distinguished configurations $c$ of embedded paths from the regular value to the critical values. In the case of the disc, we compare this family of cocycles with the Magnus cocycles given by Fox calculus and consider some abelian reductions giving rise to linear representations of braid groups. We also prove that, still in the case of the disc, the intersection numbers along straight lines, which conjecturally make sense in infinite dimensional situations, carry all the relevant information.

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