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Yusuke Kuno

Publications and source records attributed to Yusuke Kuno.

30 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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arXiv2012

Marked fatgraph complexes and surface automorphisms

Combinatorial aspects of the Torelli-Johnson-Morita theory of surface automorphisms are extended to certain subgroups of the mapping class groups. These subgroups are defined relative to a specified homomorphism from the fundamental group of the surface onto an arbitrary group $K$. For $K$ abelian, there is a combinatorial theory akin to the classical case, for example, providing an explicit cocycle representing the first Johnson homomophism with target $Λ^3 K$. Furthermore, the Earle class with coefficients in $K$ is represented by an explicit cocyle.

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arXiv2011

A combinatorial construction of symplectic expansions

The notion of a symplectic expansion directly relates the topology of a surface to formal symplectic geometry. We give a method to construct a symplectic expansion by solving a recurrence formula given in terms of the Baker-Campbell-Hausdorff series.

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arXiv2011

The generalized Dehn twist along a figure eight

For any unoriented loop on a compact connected oriented surface with one boundary component, the generalized Dehn twist along the loop is defined as an automorphism of the completed group ring of the fundamental group of the surface. If the loop is simple, this is the usual right handed Dehn twist, in particular realized as a mapping class of the surface. We investigate the case when the loop has a single transverse double point, and show that in this case the generalized Dehn twist is not realized as a mapping class.

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arXiv2011

The Meyer functions for projective varieties and their application to local signatures for fibered 4-manifolds

We study a secondary invariant, called the Meyer function, on the fundamental group of the complement of the dual variety of a smooth projective variety. This invariant have played an important role when studying the local signatures of fibered 4-manifolds from topological point of view. As an application of our study, we define a local signature for generic non-hyperelliptic fibrations of genus 4 and 5 and compute some examples.

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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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arXiv2008

A combinatorial formula for Earle's twisted 1-cocycle on the mapping class group \mathcal{M}_{g,*}

We present a formula expressing Earle's twisted 1-cocycle on the mapping class group of a closed oriented surface of genus >=2 relative to a fixed base point, with coefficients in the first homology group of the surface. For this purpose we compare it with Morita's twisted 1-cocycle which is combinatorial. The key is the computation of these cocycles on a particular element of the mapping class group, which is topologically a hyperelliptic involution.

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arXiv2008

The mapping class group and the Meyer function for plane curves

For each d>=2, the mapping class group for plane curves of degree d will be defined and it is proved that there exists uniquely the Meyer function on this group. In the case of d=4, using our Meyer function, we can define the local signature for 4-dimensional fiber spaces whose general fibers are non-hyperelliptic compact Riemann surfaces of genus 3. Some computations of our local signature will be given.

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