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Nariya Kawazumi

Publications and source records attributed to Nariya Kawazumi.

At least 19 recordsLinked to original sources
arXiv2026

$\mathbb{K}$-framings and $\mathbb{K}$-quadratic forms on surfaces

We introduce the notions of $\mathbb{K}$-framings, based $\mathbb{K}$-framings and relative $\mathbb{K}$-framings of a compact connected oriented surface $Σ$ for any commutative ring $\mathbb{K}$ with unit, and a map which maps a based loop on $Σ$ to a homology class of its unit tangent bundle $UΣ$, which recovers Johnson's lifting in the case $\mathbb{K} = \mathbb{Z}/2$. This generalizes the correspondence between a quadratic form and a spin structure established by Johnson to any commutative ring $\mathbb{K}$ with unit. If the genus of $Σ$ is positive, we have a bijection between the set of $\mathbb{K}$-framings and the set of some twisted cocycles of the mapping class group of the surface $Σ$. Through this bijection, in the case where the boundary $\partialΣ$ is non-empty and connected, we discuss some relation between $\mathbb{K}$-framings and the extended first Johnson homomorphism.

math.GT↗
arXiv2025

A topological proof of Wolpert's formula for the Weil-Petersson symplectic form in terms of the Fenchel-Nielsen coordinates

We introduce a natural cell decomposition of a closed oriented surface associated with a pants decomposition, and an explicit groupoid cocycle on the cell decomposition which represents each point of the Teichmüller space $\mathcal{T}_g$. We call it the {\it standard cocycle} of the point of $\mathcal{T}_g$. As an application of the explicit description of the standard cocycle, we obtain a topological proof of Wolpert's formula for the Weil-Petersson symplectic form in terms of the Fenchel-Nielsen coordinates associated with the pants decomposition.

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↗
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

The Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem in higher genera

For a compact oriented surface $Σ$ of genus $g$ with $n+1$ boundary components, the space $\mathfrak{g}(Σ)$ spanned by free homotopy classes of loops in $Σ$ carries the structure of a Lie bialgebra equipped with a natural decreasing filtration, whose structure morphisms are called the Goldman bracket and the (framed) Turaev cobracket. We address the following Goldman-Turaev (GT) formality problem: construct a Lie bialgebra homomorphism $θ$ from $\mathfrak{g}(Σ)$ to its associated graded ${\rm gr}\, \mathfrak{g}(Σ)$ such that ${\rm gr} \, θ= {\rm id}$. In order to solve it, we define a family of higher genus Kashiwara-Vergne (KV) problems for an element $F\in {\rm Aut}(L)$, where $L$ is a free Lie algebra. In the case of $g=0$ and $n=2$, it is the classical KV problem from Lie theory. For $g>0$, these KV problems are new. We show that an element $F$ induces a GT formality map if and only if it is a solution of the KV problem. A crucial step in solving the higher genus KV problem is to construct solutions for the case of $g=1$ and $n=1$ in terms of certain elliptic associators following Enriquez. By solving the KV problem, we establish the GT formality for every $g$ and $n$, with the exception of some framings for $g=1$ in which case the GT formality actually does not hold. Furthermore, we introduce pro-unipotent groups ${\rm KV}$ and ${\rm KRV}$ which act on the space of solutions of the KV problem freely and transitively. There are injective maps ${\rm GT}_1\to {\rm KV}, {\rm GRT}_1\to {\rm KRV}$ from Grothendieck-Teichmüller groups. As an application, we show that the Johnson obstruction given by the Turaev cobracket coincides with the one given by the Enomoto-Satoh trace. As part of our study, we prove a uniqueness theorem for non-commutative divergence cocycles on the group algebra of a free group which is of independent value.

math.GT↗
arXiv2022

A twisted invariant of a compact Riemann surface

We introduce a twisted version of the Kawazumi-Zhang invariant $a_g(C) = φ(C)$ of a compact Riemann surface $C$ of genus $g \geq 1$, and discuss how it is related to the first Mumford-Morita-Milller class $e_1 = κ_1$ on the moduli space of compact Riemann surfaces and the original Kawazumi-Zhang invariant.

math.GT↗
arXiv2021

On the wheeled PROP of stable cohomology of Aut(F_n) with bivariant coefficients

We show that the stable cohomology of automorphism groups of free groups with coefficients obtained by applying Hom(-,-) to tensor powers of the abelianization, is equipped with the structure of a wheeled PROP H. We define another wheeled PROP E by Ext-groups in the category of functors from the category of finitely generated free groups to k-modules. The main result of this paper is the construction of a morphism of wheeled PROPs $ϕ: E \to H$ such that $ϕ(E)$ is the wheeled PROP generated by the cohomology class h_1 constructed by the first author.

math.AT↗
arXiv2021

Some algebraic aspects of the Turaev cobracket

The Turaev cobracket, a loop operation introduced by V. Turaev, which measures self-intersection of a loop on a surface, is a modification of a path operation introduced earlier by Turaev himself, as well as a counterpart of the Goldman bracket. In this survey based on the author's joint works with A. Alekseev, Y. Kuno and F. Naef, we review some algebraic aspects of the cobracket and its framed variants including their formal description, an application to the mapping class group of the surface and a relation to the (higher genus) Kashiwara-Vergne problem. In addition, we review a homological description of the cobracket after R. Hain.

math.GT↗
arXiv2018

Goldman-Turaev formality implies Kashiwara-Vergne

Let $Σ$ be a compact connected oriented 2-dimensional manifold with non-empty boundary. In our previous work, we have shown that the solution of generalized (higher genus) Kashiwara-Vergne equations for an automorphism $F \in {\rm Aut}(L)$ of a free Lie algebra implies an isomorphism between the Goldman-Turaev Lie bialgebra $\mathfrak{g}(Σ)$ and its associated graded ${\rm gr}\, \mathfrak{g}(Σ)$. In this paper, we prove the converse: if $F$ induces an isomorphism $\mathfrak{g}(Σ) \cong {\rm gr} \, \mathfrak{g}(Σ)$, then it satisfies the Kashiwara-Vergne equations up to conjugation. As an application of our results, we compute the degree one non-commutative Poisson cohomology of the Kirillov-Kostant-Souriau double bracket. The main technical tool used in the paper is a novel characterization of conjugacy classes in the free Lie algebra in terms of cyclic words.

math.GT↗
arXiv2018

The Goldman-Turaev Lie bialgebra in genus zero and the Kashiwara-Vergne problem

In this paper, we describe a surprising link between the theory of the Goldman-Turaev Lie bialgebra on surfaces of genus zero and the Kashiwara-Vergne (KV) problem in Lie theory. Let $Σ$ be an oriented 2-dimensional manifold with non-empty boundary and $\mathbb{K}$ a field of characteristic zero. The Goldman-Turaev Lie bialgebra is defined by the Goldman bracket $\{ -,- \}$ and Turaev cobracket $δ$ on the $\mathbb{K}$-span of homotopy classes of free loops on $Σ$. Applying an expansion $θ: \mathbb{K}π\to \mathbb{K}\langle x_1, \dots, x_n \rangle$ yields an algebraic description of the operations $\{ -,- \}$ and $δ$ in terms of non-commutative variables $x_1, \dots, x_n$. If $Σ$ is a surface of genus $g=0$ the lowest degree parts $\{ -,- \}_{-1}$ and $δ_{-1}$ are canonically defined (and independent of $θ$). They define a Lie bialgebra structure on the space of cyclic words which was introduced and studied by T. Schedler. It was conjectured by the second and the third authors that one can define an expansion $θ$ such that $\{ -,- \}=\{ -,- \}_{-1}$ and $δ=δ_{-1}$. The main result of this paper states that for surfaces of genus zero constructing such an expansion is essentially equivalent to the KV problem. G. Massuyeau constructed such expansions using the Kontsevich integral. In order to prove this result, we show that the Turaev cobracket $δ$ can be constructed in terms of the double bracket (upgrading the Goldman bracket) and the non-commutative divergence cocycle which plays the central role in the KV theory. Among other things, this observation gives a new topological interpretation of the KV problem and allows to extend it to surfaces with arbitrary number of boundary components (and of arbitrary genus, see [C. R. Acad. Sci. Paris, Ser. I 355 (2017), 123--127]).

math.GT↗
arXiv2017

The mapping class group orbits in the framings of compact surfaces

We compute the mapping class group orbits in the homotopy set of framings of a compact connected oriented surface with non-empty boundary. In the case $g > 1$ the computation is some modification of Johnson's results and certain arguments on the Arf invariant, while we need an extra invariant for the genus $1$ case. In addition, we discuss how this invariant behaves in the relative case, which Randal-Williams studied for $g > 1$.

math.GT↗
arXiv2016

Higher genus Kashiwara-Vergne problems and the Goldman-Turaev Lie bialgebra

We define a family ${\rm KV}^{(g,n)}$ of Kashiwara-Vergne problems associated with compact connected oriented 2-manifolds of genus $g$ with $n+1$ boundary components. The problem ${\rm KV}^{(0,3)}$ is the classical Kashiwara-Vergne problem from Lie theory. We show the existence of solutions of ${\rm KV}^{(g,n)}$ for arbitrary $g$ and $n$. The key point is the solution of ${\rm KV}^{(1,1)}$ based on the results by B. Enriquez on elliptic associators. Our construction is motivated by applications to the formality problem for the Goldman-Turaev Lie bialgebra $\mathfrak{g}^{(g, n+1)}$. In more detail, we show that every solution of ${\rm KV}^{(g,n)}$ induces a Lie bialgebra isomorphism between $\mathfrak{g}^{(g, n+1)}$ and its associated graded ${\rm gr} \, \mathfrak{g}^{(g, n+1)}$. For $g=0$, a similar result was obtained by G. Massuyeau using the Kontsevich integral. This paper is a summary of our results. Details and proofs will appear elsewhere.

math.QA↗
arXiv2015

Generalized Kronecker formula for Bernoulli numbers and self-intersections of curves on a surface

We present a new explicit formula for the $m$-th Bernoulli number $B_m$, which involves two integer parameters $a$ and $n$ with $0\le a\le m\le n$. If we set $a=0$ and $n=m$, then the formula reduces to the celebrated Kronecker formula for $B_m$. We give two proofs of our formula. One is analytic and uses a certain function in two variables. The other is algebraic and is motivated by a topological consideration of self-intersections of curves on an oriented surface.

math.NT↗
arXiv2013

Surface topology and involutive bimodules

We remark some basic facts on homological aspects of involutive Lie bialgebras and their involutive bimodules, and present some problems on surface topology related to these facts.

math.GT↗