Online seminar on
Johnson homomorphisms and related topics
As a continuation of the previous workshops "Johnson homomorphisms and related topics" held in 2013, 2017 and 2019, we launch this online seminar. The scope of this seminar includes not only the Johnson homomorphisms for the mapping class groups, but also all related topics in topology and algebra, such as invariants of 3-manifolds, automorphism groups of free groups, graph homology, arithmetic or motivic creations, etc. Let us share our ideas on these fruitful topics spreading from the Johnson homomorphisms!
Organizers:
Nariya Kawazumi, Gwénaël Massuyeau, Hiroaki Nakamura, Takuya Sakasai, Christine Vespa.
Details
The seminar uses Zoom and the talks are given in English.
All participants are requested to have a pre-registration for each seminar.
The seminar will be held in the morning in Europe and the afternoon in East Asia as is displayed in the program below.
Program
February 7th (Mon), 2022.
Pre-registration (click here) is required.
9:15-10:00 (CET, France)=17:15-18:00 (JST, Japan)
Oscar Randal-Williams (University of Cambridge)
The Torelli Lie algebra
Abstract: The Mal'cev Lie algebra associated to the Torelli group of a surface was completely determined by Hain (1993), who gave an explicit presentation which is quadratic if the genus is at least 4. I will explain some work, joint with A. Kupers, which exploits the formal similarity between surfaces and certain higher-dimensional manifolds to prove some new results about this Lie algebra: it is stably Koszul, and the geometric Johnson homomorphism is (nearly) stably injective.
10:15-11:00 (CET, France)=18:15-19:00 (JST, Japan)
Mai Katada (Kyoto University)
Actions of automorphism groups of free groups on spaces of Jacobi diagrams
Abstract: We construct a polynomial functor Ad of degree 2d from the opposite category of the category of finitely generated free groups to the category of filtered vector spaces. The functor Ad induces an action of the automorphism group Aut(Fn) of the free group Fn of rank n on a filtered vector space Ad(n) consisting of Jacobi diagrams of degree d on n oriented arcs. This Aut(Fn)-action induces two actions on the associated graded vector space of Ad(n), which is identified with the space Bd(n) of colored open Jacobi diagrams: a GL(n,Z)-action and an action of the graded Lie algebra gr(IA(n)) associated with the lower central series of the IA-automorphism group of Fn. We use these two actions on Bd(n) to study the Aut(Fn)-module structure of Ad(n). We obtain for n≧2d, an indecomposable decomposition of Ad(n) and the radical filtration of Ad(n).
Previous seminars
October 6th (Wed), 2021.
10:15-11:00 (CEST, France)=17:15-18:00 (JST, Japan)
Christian Blanchet (University of Paris)
Representations of Mapping Class Groups from Heisenberg quotient of surface braid groups
Abstract: Motivated by the famous Lawrence–Krammer–Bigelow representations of the classical braid groups, we study homology of unordered configuration spaces of a surface with one boundary component, over a local system which is defined from a Heisenberg quotient of the surface braid group.
In general we obtain a representation of the Chillingworth subgroup, a representation of a central extension of the Torelli subgroup and a twisted representation of the whole Mapping Class Group.
By specialising to the Schrodinger representation, or its finite dimensional analogue at a root of unity, we obtain representations of the metaplectic cover of the Mapping Class Group.
This is joint work with Martin Palmer and Awais Shaukat, arXiv:2109.00515.
11:15-12:00 (CEST, France)=18:15-19:00 (JST, Japan)
Yuta Nozaki (Hiroshima University)
On the kernel of the surgery map restricted to the 1-loop part
Abstract: Claspers in 3-manifolds enable us to define the surgery map from the module of Jacobi diagrams to the graded quotient of the Y-filtration of the monoid of homology cylinders.
We determine the kernel of the surgery map restricted to the 1-loop part after taking a certain quotient of the target.
The key tools are a homomorphism defined via the LMO functor and a refinement of the surgery map.
This is joint work with Masatoshi Sato and Masaaki Suzuki.
November 23rd (Tue), 2021.
9:15-10:00 (CET, France)=17:15-18:00 (JST, Japan)
Wolfgang Pitsch (Universitat Autònoma de Barcelona)
Rational homology spheres and trivial cocycles
Abstract: In this talk I will explain our results with my former PhD student Ricard Riba. I will first show how to extend Singer's theorem relating double mapping class group classes and oriented manifolds to the case of rational homology spheres. This ties the study of the set of rational homology spheres to that of the mod-p Torelli subgroups of the mapping class group, i.e. the kernels of the canonical map onto the symplectic groups with mod p coefficients. I will then explain how this allows us to generalize our results relating invariants of integral homology spheres and trivial cocycles on the Torelli group to this setting. Time permits I will explain how to use this framework to disprove a conjecture from Perron on a tentative extension of the mod-p Casson invariant to the set of rational homology spheres.
10:15-11:00 (CET, France)=18:15-19:00 (JST, Japan)
Erik Lindell (Stockholm University)
Abelian cycles in the homology of the Torelli group
Abstract: We study the stable rational homology of the Torelli group of a compact and orientable surface of genus g with a boundary component or a marked point, using the Sp_{2g}(Z)-equivariant map induced on homology by the Johnson homomorphism. The image of this map is a finite dimensional and algebraic Sp_{2g}(Z)-representation. By considering a class of homology classes called abelian cycles, which are easy to write down and for which we have an explicit formula for the map in question, we can use classical representation theory of symplectic groups to describe a large part of this image.
January 12th (Wed), 2022.
9:15-10:00 (CET, France)=17:15-18:00 (JST, Japan)
Hidekazu Furusho (Nagoya University)
Associators and an l-adic analogue of Gauss's hypergeometric function
Abstract: I will talk about a construction of an l-adic analogue of Gauss's hypergeometric function.
It arises from an l-adic Galois representation on a fundamental torsor of the projective line minus three points and a comparison isomorphism between Betti and de Rham fundamental groups constructed by an even associator. Its definition is motivated by a relation between the KZ-equation and the hypergeometric function in the complex case. I plan to explain two basic properties, analogues of Gauss's hypergeometric theorem and that of Euler's transformation formula for the l-adic function.
The proofs are based on a connection of certain two-by-two matrices specializations of even associators with the associated gamma functions, which extends the result of Ohno and Zagier.
10:15-11:00 (CET, France)=18:15-19:00 (JST, Japan)
Jacques Darné (Université catholique de Louvain)
Lower central series of partitioned braids on surfaces
Abstract: Partitioned braid groups (sometimes called "mixed braid groups") are subgroups of the braid group standing between the pure braid group $P_n$ and the whole braid group $B_n$. On the one hand, the lower central series of $B_n$ is almost trivial. On the other hand, the lower central series of $P_n$ is a very rich object, encoding finite type invariants of braids. As a consequence, one can expect partitioned braid groups to display a range of intermediate behaviors, and this is indeed what we observe. In this talk, we will explore these different behaviours and give an answer to the first question one can ask about these lower central series : when do they stop ? Even this simple question turns out to be a difficult one, especially when one considers its generalization to braids on surfaces. However, we will be able to answer it almost completely, leaving open only some cases of partitioned braids on the projective plane.