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Nariya Kawazumi

Publications and source records attributed to Nariya Kawazumi.

31 records · Page 2Linked to original sources
arXiv2012

Groupoid-theoretical methods in the mapping class groups of surfaces

We provide some language for algebraic study of the mapping class groups for surfaces with non-connected boundary. As applications, we generalize our previous results on Dehn twists to any compact connected oriented surfaces with non-empty boundary. Moreover we embed the `smallest' Torelli group in the sense of Putman into a pro-nilpotent group coming from the Goldman Lie algebra. The graded quotients of the embedding equal the Johnson homomorphisms of all degrees if the boundary is connected.

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arXiv2012

The center of the Goldman Lie algebra of a surface of infinite genus

Let $Σ_{\infty, 1}$ be the inductive limit of compact oriented surfaces with one boundary component. We prove the center of the Goldman Lie algebra of the surface $Σ_{\infty,1}$ is spanned by the constant loop. A similar statement for a closed oriented surface was conjectured by Chas and Sullivan, and proved by Etingof. Our result is deduced from a computation of the center of the Lie algebra of oriented chord diagrams.

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arXiv2011

The Lie algebra of rooted planar trees

We study a natural Lie algebra structure on the free vector space generated by all rooted planar trees as the associated Lie algebra of the nonsymmetric operad (non-$Σ$ operad, preoperad) of rooted planar trees. We determine whether the Lie algebra and some related Lie algebras are finitely generated or not, and prove that a natural surjection called the augmentation homomorphism onto the Lie algebra of polynomial vector fields on the line has no splitting preserving the units.

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arXiv2010

The logarithms of Dehn twists

By introducing an invariant of loops on a compact oriented surface with one boundary component, we give an explicit formula for the action of Dehn twists on the completed group ring of the fundamental group of the surface. This invariant can be considered as ``the logarithms" of Dehn twists. The formula generalizes the classical formula describing the action on the first homology of the surface, and Morita's explicit computations of the extended first and the second Johnson homomorphisms. For the proof we use a homological interpretation of the Goldman Lie algebra in the framework of Kontsevich's formal symplectic geometry. As an application, we prove the action of the Dehn twist of a simple closed curve on the $k$-th nilpotent quotient of the fundamental group of the surface depends only on the conjugacy class of the curve in the $k$-th quotient.

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arXiv2009

Twisted Morita-Mumford classes on braid groups

Evaluating the twisted Morita-Mumford classes bar(h)_p on the Artin braid group B_n, we give the stable algebraic independence of the bar(h)_p's on the automorphism group of the free group, Aut(F_n). This is sharper than the results we obtained by restricting them to the mapping class group.

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arXiv2008

Harmonic Magnus Expansion on the Universal Family of Riemann Surfaces

Let ${\mathbb M}_{g, 1}$, $g \geq 1$, be the moduli space of triples $(C, P_0, v)$ of genus $g$, where $C$ is a compact Riemann surface of genus $g$, $P_0 \in C$, and $v \in T_{P_0}C\setminus\{0\}$. Using Chen's iterated integrals we introduce a higher analogue of the period matrix for a triple $(C, P_0, v)$, {\it the harmonic Magnus expansion}. It induces a flat connection on a vector bundle over the space ${\mathbb M}_{g, 1}$, whose holonomy gives all the higher Johnson homomorphisms of the mapping class group. The connection form, which is computed as an explicit quadratic differential, induces "canonical" differential forms representing (twisted) Morita-Mumford classes and their higher relators on ${\mathbb M}_{g, 1}$. In particular, we construct a family of twisted differential forms on ${\mathbb M}_{g, 1}$ representing the $(0, p+2)$-twisted Morita-Mumford class $m_{0, p+2}$ combinatorially parametrized by the Stasheff associahedron $K_{p+1}$.

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arXiv2008

Johnson's homomorphisms and the Arakelov-Green function

Let $π: {\mathbb C}_g \to {\mathbb M}_g$ be the universal family of compact Riemann surfaces of genus $g \geq 1$. We introduce a real-valued function on the moduli space ${\mathbb M}_g$ and compute the first and the second variations of the function. As a consequence we relate the Chern form of the relative tangent bundle $T_{{\mathbb C}_g/{\mathbb M}_g}$ induced by the Arakelov-Green function with differential forms on ${\mathbb C}_g$ induced by a flat connection whose holonomy gives Johnson's homomorphisms on the mapping class group.

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arXiv2007

Canonical 2-forms on the moduli space of Riemann surfaces

As was shown by Harer the second homology of ${\mathbb M}_g$, the moduli space of compact Riemann surfaces of genus $g$, is of rank 1, provided $g \geq 3$. This means a nontrivial second de Rham cohomology class on ${\mathbb M}_g$ is unique up to constant factor. But several canonical 2-forms on the moduli space have been constructed in various geometric contexts, and differ from each other. In this article we review some of constructions in order to provide materials for future research on "secondary geometry" of the moduli space ${\mathbb M}_g$.

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arXiv2007

Canonical lifts of the Johnson homomorphisms to the Torelli groupoid

We prove that every trivalent marked bordered fatgraph comes equipped with a canonical generalized Magnus expansion in the sense of Kawazumi. This Magnus expansion is used to give canonical lifts of the higher Johnson homomorphisms $τ_m$, for $m\geq 1$, to the Torelli groupoid, and we provide a recursive combinatorial formula for tensor representatives of these lifts. In particular, we give an explicit 1-cocycle in the dual fatgraph complex which lifts $τ_2$ and thus answer affirmatively a question of Morita-Penner. To illustrate our techniques for calculating higher Johnson homomorphisms in general, we give explicit examples calculating $τ_m$, for $m\leq 3$.

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arXiv2006

Cohomological Aspects of Magnus Expansions

We generalize the notion of a Magnus expansion of a free group in order to extend each of the Johnson homomorphisms defined on a decreasing filtration of the Torelli group for a surface with one boundary component to the whole of the automorphism group of a free group $\operatorname{Aut}(F_{n})$. The extended ones are {\it not} homomorphisms, but satisfy an infinite sequence of coboundary relations, so that we call them {\it the Johnson maps}. In this paper we confine ourselves to studying the first and the second relations, which have cohomological consequences about the group $\operatorname{Aut}(F_{n})$ and the mapping class groups for surfaces. The first one means that the first Johnson map is a twisted 1-cocycle of the group $\operatorname{Aut}(F_{n})$. Its cohomology class coincides with ``the unique elementary particle" of all the Morita-Mumford classes on the mapping class group for a surface [Ka1] [KM1]. The second one restricted to the mapping class group is equal to a fundamental relation among twisted Morita-Mumford classes proposed by Garoufalidis and Nakamura [GN] and established by Morita and the author [KM2]. This means we give a simple and coherent proof of the fundamental relation. The first Johnson map gives the abelianization of the induced automorphism group $IA_n$ of a free group in an explicit way.

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arXiv2005

An application of the second Riemann continuation theorem to cohomology of the Lie algebra of vector fields on the complex line

We study cohomology groups of the Lie algebra of vector fields on the complex line, $W_1$, with values in the tensor fields in several variables. From a generalization by Scheja of the second Riemann (Hartogs) continuation theorem, we deduce a cohomology exact sequence of the subalgebra of $W_1$ consisting of vectors having a zero at the origin. As applications, we compute the cohomology algebra of $W_1$ with values in the functions on $\Bbb C^n$ explicitly, and establish a certain vanishing theorem for the cohomology of $W_1$ with values in the quadratic differentials in several variables, which is closely related to the moduli space of Riemann surfaces.

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