## Tuesday Seminar on Topology

Seminar information archive ～04/15｜Next seminar｜Future seminars 04/16～

Date, time & place | Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya |

**Seminar information archive**

### 2014/01/21

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Weak eigenvalues in homoclinic classes: perturbations from saddles

with small angles (ENGLISH)

**Xiaolong Li**(The University of Tokyo)Weak eigenvalues in homoclinic classes: perturbations from saddles

with small angles (ENGLISH)

[ Abstract ]

For 3-dimensional homoclinic classes of saddles with index 2, a

new sufficient condition for creating weak contracting eigenvalues is

provided. Our perturbation makes use of small angles between stable and

unstable subspaces of saddles. In particular, by recovering the unstable

eigenvector, we can designate that the newly created weak eigenvalue is

contracting. As applications, we obtain C^1-generic non-trivial index-

intervals of homoclinic classes and the C^1-approximation of robust

heterodimensional cycles. In particular, this sufficient condition is

satisfied by a substantial class of saddles with homoclinic tangencies.

For 3-dimensional homoclinic classes of saddles with index 2, a

new sufficient condition for creating weak contracting eigenvalues is

provided. Our perturbation makes use of small angles between stable and

unstable subspaces of saddles. In particular, by recovering the unstable

eigenvector, we can designate that the newly created weak eigenvalue is

contracting. As applications, we obtain C^1-generic non-trivial index-

intervals of homoclinic classes and the C^1-approximation of robust

heterodimensional cycles. In particular, this sufficient condition is

satisfied by a substantial class of saddles with homoclinic tangencies.

### 2014/01/14

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

State-integral partition functions on shaped triangulations (ENGLISH)

**Rinat Kashaev**(University of Geneva)State-integral partition functions on shaped triangulations (ENGLISH)

[ Abstract ]

Quantum Teichm\\"uller theory can be promoted to a

generalized TQFT within the combinatorial framework of shaped

triangulations with the tetrahedral weight functions given in

terms of the Weil-Gelfand-Zak transformation of Faddeev.FN"s

quantum dilogarithm. By using simple examples, I will

illustrate the connection of this theory with the hyperbolic

geometry in three dimensions.

Quantum Teichm\\"uller theory can be promoted to a

generalized TQFT within the combinatorial framework of shaped

triangulations with the tetrahedral weight functions given in

terms of the Weil-Gelfand-Zak transformation of Faddeev.FN"s

quantum dilogarithm. By using simple examples, I will

illustrate the connection of this theory with the hyperbolic

geometry in three dimensions.

### 2013/12/24

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Stable homotopy type for monopole Floer homology (ENGLISH)

**Tirasan Khandhawit**(Kavli IPMU)Stable homotopy type for monopole Floer homology (ENGLISH)

[ Abstract ]

In this talk, I will try to give an overview of the

construction of stable homotopy type for monopole Floer homology. The

construction associates a stable homotopy object to 3-manifolds, which

will recover the Floer groups by appropriate homology theory. The main

ingredients are finite dimensional approximation technique and Conley

index theory. In addition, I will demonstrate construction for certain

3-manifolds such as the 3-torus.

In this talk, I will try to give an overview of the

construction of stable homotopy type for monopole Floer homology. The

construction associates a stable homotopy object to 3-manifolds, which

will recover the Floer groups by appropriate homology theory. The main

ingredients are finite dimensional approximation technique and Conley

index theory. In addition, I will demonstrate construction for certain

3-manifolds such as the 3-torus.

### 2013/12/17

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Satellites of an oriented surface link and their local moves (JAPANESE)

**Inasa Nakamura**(The University of Tokyo)Satellites of an oriented surface link and their local moves (JAPANESE)

[ Abstract ]

For an oriented surface link $F$ in $\\mathbb{R}^4$,

we consider a satellite construction of a surface link, called a

2-dimensional braid over $F$, which is in the form of a covering over

$F$. We introduce the notion of an $m$-chart on a surface diagram

$p(F)\\subset \\mathbb{R}^3$ of $F$, which is a finite graph on $p(F)$

satisfying certain conditions and is an extended notion of an

$m$-chart on a 2-disk presenting a surface braid.

A 2-dimensional braid over $F$ is presented by an $m$-chart on $p(F)$.

It is known that two surface links are equivalent if and only if their

surface diagrams are related by a finite sequence of ambient isotopies

of $\\mathbb{R}^3$ and local moves called Roseman moves.

We show that Roseman moves for surface diagrams with $m$-charts can be

well-defined. Further, we give some applications.

For an oriented surface link $F$ in $\\mathbb{R}^4$,

we consider a satellite construction of a surface link, called a

2-dimensional braid over $F$, which is in the form of a covering over

$F$. We introduce the notion of an $m$-chart on a surface diagram

$p(F)\\subset \\mathbb{R}^3$ of $F$, which is a finite graph on $p(F)$

satisfying certain conditions and is an extended notion of an

$m$-chart on a 2-disk presenting a surface braid.

A 2-dimensional braid over $F$ is presented by an $m$-chart on $p(F)$.

It is known that two surface links are equivalent if and only if their

surface diagrams are related by a finite sequence of ambient isotopies

of $\\mathbb{R}^3$ and local moves called Roseman moves.

We show that Roseman moves for surface diagrams with $m$-charts can be

well-defined. Further, we give some applications.

### 2013/12/10

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Corks, plugs, and local moves of 4-manifolds. (JAPANESE)

**Motoo Tange**(University of Tsukuba)Corks, plugs, and local moves of 4-manifolds. (JAPANESE)

[ Abstract ]

Akbulut and Yasui defined cork, and plug

to produce many exotic pairs.

In this talk, we introduce a plug

with respect to Fintushel-Stern's knot surgery

or more 4-dimensional local moves and

and argue by using Heegaard Fleor theory.

Akbulut and Yasui defined cork, and plug

to produce many exotic pairs.

In this talk, we introduce a plug

with respect to Fintushel-Stern's knot surgery

or more 4-dimensional local moves and

and argue by using Heegaard Fleor theory.

### 2013/12/03

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Hyperbolic four-manifolds with one cusp (cancelled) (JAPANESE)

**Bruno Martelli**(Univ. di Pisa)Hyperbolic four-manifolds with one cusp (cancelled) (JAPANESE)

[ Abstract ]

(joint work with A. Kolpakov)

We introduce a simple algorithm which transforms every

four-dimensional cubulation into a cusped finite-volume hyperbolic

four-manifold. Combinatorially distinct cubulations give rise to

topologically distinct manifolds. Using this algorithm we construct

the first examples of finite-volume hyperbolic four-manifolds with one

cusp. More generally, we show that the number of k-cusped hyperbolic

four-manifolds with volume smaller than V grows like C^{V log V} for

any fixed k. As a corollary, we deduce that the 3-torus bounds

geometrically a hyperbolic manifold.

(joint work with A. Kolpakov)

We introduce a simple algorithm which transforms every

four-dimensional cubulation into a cusped finite-volume hyperbolic

four-manifold. Combinatorially distinct cubulations give rise to

topologically distinct manifolds. Using this algorithm we construct

the first examples of finite-volume hyperbolic four-manifolds with one

cusp. More generally, we show that the number of k-cusped hyperbolic

four-manifolds with volume smaller than V grows like C^{V log V} for

any fixed k. As a corollary, we deduce that the 3-torus bounds

geometrically a hyperbolic manifold.

### 2013/11/26

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Rational elliptic surfaces and certain line-conic arrangements (JAPANESE)

**Hiroo Tokunaga**(Tokyo Metropolitan University)Rational elliptic surfaces and certain line-conic arrangements (JAPANESE)

[ Abstract ]

Let S be a rational elliptic surface. The generic

fiber of S can be considered as an elliptic curve over

the rational function field of one variable. We can make

use of its group structure in order to cook up a curve C_2 on

S from a given section C_1.

In this talk, we consider certain line-conic arrangements of

degree 7 based on this method.

Let S be a rational elliptic surface. The generic

fiber of S can be considered as an elliptic curve over

the rational function field of one variable. We can make

use of its group structure in order to cook up a curve C_2 on

S from a given section C_1.

In this talk, we consider certain line-conic arrangements of

degree 7 based on this method.

### 2013/11/19

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Minimal $C^1$-diffeomorphisms of the circle which admit

measurable fundamental domains (JAPANESE)

**Hiroki Kodama**(The University of Tokyo)Minimal $C^1$-diffeomorphisms of the circle which admit

measurable fundamental domains (JAPANESE)

[ Abstract ]

We construct, for each irrational number $\\alpha$, a minimal

$C^1$-diffeomorphism of the circle with rotation number $\\alpha$

which admits a measurable fundamental domain with respect to

the Lebesgue measure.

This is a joint work with Shigenori Matsumoto (Nihon University).

We construct, for each irrational number $\\alpha$, a minimal

$C^1$-diffeomorphism of the circle with rotation number $\\alpha$

which admits a measurable fundamental domain with respect to

the Lebesgue measure.

This is a joint work with Shigenori Matsumoto (Nihon University).

### 2013/11/12

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

The Batalin-Vilkovisky Formalism and Cohomology of Moduli Spaces (ENGLISH)

**Alexander Voronov**(University of Minnesota)The Batalin-Vilkovisky Formalism and Cohomology of Moduli Spaces (ENGLISH)

[ Abstract ]

We use the Batalin-Vilkovisky formalism to give a new proof of Costello's theorem on the existence and uniqueness of solution to the Quantum Master Equation. We also make a physically motivated conjecture on the rational homology of moduli spaces. This is a joint work with Domenico D'Alessandro.

We use the Batalin-Vilkovisky formalism to give a new proof of Costello's theorem on the existence and uniqueness of solution to the Quantum Master Equation. We also make a physically motivated conjecture on the rational homology of moduli spaces. This is a joint work with Domenico D'Alessandro.

### 2013/11/05

16:30-18:00 Room #123 (Graduate School of Math. Sci. Bldg.)

The isotopy problem of non-singular closed 1-forms. (ENGLISH)

**Carlos Moraga Ferrandiz**(The University of Tokyo, JSPS)The isotopy problem of non-singular closed 1-forms. (ENGLISH)

[ Abstract ]

Given alpha_0, alpha_1 two cohomologous non-singular closed 1-forms of a compact manifold M, are they always isotopic? We expect a negative answer to this question, at least in high dimensions by the work of Laudenbach, as well as an obstruction living in the algebraic K-theory of the Novikov ring associated to the underlying cohomology class.

A similar problem for functions N x [0,1] --> [0,1] without critical points was treated by Hatcher and Wagoner in the 70s.

The first goal of this talk is to explain how we can carry a part of the strategy of Hatcher and Wagoner into the context of closed 1-forms and to indicate the main difficulties that appear by doing so. The second goal is to show the techniques to treat this difficulties and the progress in defining the expected obstruction.

Given alpha_0, alpha_1 two cohomologous non-singular closed 1-forms of a compact manifold M, are they always isotopic? We expect a negative answer to this question, at least in high dimensions by the work of Laudenbach, as well as an obstruction living in the algebraic K-theory of the Novikov ring associated to the underlying cohomology class.

A similar problem for functions N x [0,1] --> [0,1] without critical points was treated by Hatcher and Wagoner in the 70s.

The first goal of this talk is to explain how we can carry a part of the strategy of Hatcher and Wagoner into the context of closed 1-forms and to indicate the main difficulties that appear by doing so. The second goal is to show the techniques to treat this difficulties and the progress in defining the expected obstruction.

### 2013/10/29

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Fundamental groups of algebraic varieties (ENGLISH)

**Daniel Matei**(IMAR, Bucharest)Fundamental groups of algebraic varieties (ENGLISH)

[ Abstract ]

We discuss restrictions imposed by the complex

structure on fundamental groups of quasi-projective

algebraic varieties with mild singularities.

We investigate quasi-projectivity of various geometric

classes of finitely presented groups.

We discuss restrictions imposed by the complex

structure on fundamental groups of quasi-projective

algebraic varieties with mild singularities.

We investigate quasi-projectivity of various geometric

classes of finitely presented groups.

### 2013/10/22

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Cluster algebra and complex volume of knots (JAPANESE)

**Rei Inoue**(Chiba University)Cluster algebra and complex volume of knots (JAPANESE)

[ Abstract ]

The cluster algebra was introduced by Fomin and Zelevinsky around

2000. The characteristic operation in the algebra called `mutation' is

related to various notions in mathematics and mathematical physics. In

this talk I review a basics of the cluster algebra, and introduce its

application to study the complex volume of knot complements in S^3.

Here a mutation corresponds to an ideal tetrahedron.

This talk is based on joint work with Kazuhiro Hikami (Kyushu University).

The cluster algebra was introduced by Fomin and Zelevinsky around

2000. The characteristic operation in the algebra called `mutation' is

related to various notions in mathematics and mathematical physics. In

this talk I review a basics of the cluster algebra, and introduce its

application to study the complex volume of knot complements in S^3.

Here a mutation corresponds to an ideal tetrahedron.

This talk is based on joint work with Kazuhiro Hikami (Kyushu University).

### 2013/10/15

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Desingularizing special generic maps (JAPANESE)

**Masamichi Takase**(Seikei University)Desingularizing special generic maps (JAPANESE)

[ Abstract ]

This is a joint work with Osamu Saeki (IMI, Kyushu University).

A special generic map is a generic map which has only definite

fold as its singularities.

We study the condition for a special generic map from a closed

n-manifold to the p-space (n+1>p), to factor through a codimension

one immersion (or an embedding). In particular, for the cases

where p = 1 and 2 we obtain complete results.

Our techniques are related to Smale-Hirsch theory,

topology of the space of immersions, relation between the space

of topological immersions and that of smooth immersions,

sphere eversions, differentiable structures of homotopy spheres,

diffeomorphism group of spheres, free group actions on the sphere, etc.

This is a joint work with Osamu Saeki (IMI, Kyushu University).

A special generic map is a generic map which has only definite

fold as its singularities.

We study the condition for a special generic map from a closed

n-manifold to the p-space (n+1>p), to factor through a codimension

one immersion (or an embedding). In particular, for the cases

where p = 1 and 2 we obtain complete results.

Our techniques are related to Smale-Hirsch theory,

topology of the space of immersions, relation between the space

of topological immersions and that of smooth immersions,

sphere eversions, differentiable structures of homotopy spheres,

diffeomorphism group of spheres, free group actions on the sphere, etc.

### 2013/10/08

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

An invariant of rational homology 3-spheres via vector fields. (JAPANESE)

**Tatsuro Shimizu**(The Univesity of Tokyo)An invariant of rational homology 3-spheres via vector fields. (JAPANESE)

[ Abstract ]

In this talk, we define an invariant of rational homology 3-spheres with

values in a space $\\mathcal A(\\emptyset)$ of Jacobi diagrams by using

vector fields.

The construction of our invariant is a generalization of both that of

the Kontsevich-Kuperberg-Thurston invariant $z^{KKT}$

and that of Fukaya and Watanabe's Morse homotopy invariant $z^{FW}$.

As an application of our invariant, we prove that $z^{KKT}=z^{FW}$ for

integral homology 3-spheres.

In this talk, we define an invariant of rational homology 3-spheres with

values in a space $\\mathcal A(\\emptyset)$ of Jacobi diagrams by using

vector fields.

The construction of our invariant is a generalization of both that of

the Kontsevich-Kuperberg-Thurston invariant $z^{KKT}$

and that of Fukaya and Watanabe's Morse homotopy invariant $z^{FW}$.

As an application of our invariant, we prove that $z^{KKT}=z^{FW}$ for

integral homology 3-spheres.

### 2013/10/01

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

The geography problem of Lefschetz fibrations (JAPANESE)

**Naoyuki Monden**(Tokyo University of Science)The geography problem of Lefschetz fibrations (JAPANESE)

[ Abstract ]

To consider holomorphic fibrations complex surfaces over complex curves

and Lefschetz fibrations over surfaces is one method for the study of

complex surfaces of general type and symplectic 4-manifods, respectively.

In this talk, by comparing the geography problem of relatively minimal

holomorphic fibrations with that of relatively minimal Lefschetz

fibrations (i.e., the characterization of pairs $(x,y)$ of certain

invariants $x$ and $y$ corresponding to relatively minimal holomorphic

fibrations and relatively minimal Lefschetz fibrations), we observe the

difference between complex surfaces of general type and symplectic

4-manifolds. In particular, we construct Lefschetz fibrations violating

the ``slope inequality" which holds for any relatively minimal holomorphic

fibrations.

To consider holomorphic fibrations complex surfaces over complex curves

and Lefschetz fibrations over surfaces is one method for the study of

complex surfaces of general type and symplectic 4-manifods, respectively.

In this talk, by comparing the geography problem of relatively minimal

holomorphic fibrations with that of relatively minimal Lefschetz

fibrations (i.e., the characterization of pairs $(x,y)$ of certain

invariants $x$ and $y$ corresponding to relatively minimal holomorphic

fibrations and relatively minimal Lefschetz fibrations), we observe the

difference between complex surfaces of general type and symplectic

4-manifolds. In particular, we construct Lefschetz fibrations violating

the ``slope inequality" which holds for any relatively minimal holomorphic

fibrations.

### 2013/07/16

17:10-18:10 Room #056 (Graduate School of Math. Sci. Bldg.)

On new models of real hyperbolic spaces (JAPANESE)

**Sumio Yamada**(Gakushuin University)On new models of real hyperbolic spaces (JAPANESE)

[ Abstract ]

In this talk, I will introduce several new realization of the real hyperbolic spaces, using classical tools. The constructions will involve aspects of convex geometry as well as projective geometry, and they are interesting from the view point of the history of mathematics. This work belongs to a joint project with Athanase Papadopoulos.

In this talk, I will introduce several new realization of the real hyperbolic spaces, using classical tools. The constructions will involve aspects of convex geometry as well as projective geometry, and they are interesting from the view point of the history of mathematics. This work belongs to a joint project with Athanase Papadopoulos.

### 2013/07/09

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Smooth 3-manifolds in the 4-sphere (ENGLISH)

**Ryan Budney**(University of Victoria)Smooth 3-manifolds in the 4-sphere (ENGLISH)

[ Abstract ]

Everyone who has studied topology knows the compact 2-manifolds that embed in the 3-sphere. One dimension up, the problem of which smooth 3-manifolds embed in the 4-sphere turns out to be much more involved with a handful of partial answers. I will describe what is known at the present moment.

Everyone who has studied topology knows the compact 2-manifolds that embed in the 3-sphere. One dimension up, the problem of which smooth 3-manifolds embed in the 4-sphere turns out to be much more involved with a handful of partial answers. I will describe what is known at the present moment.

### 2013/06/25

17:10-18:10 Room #056 (Graduate School of Math. Sci. Bldg.)

Higher-order generalization of Fukaya's Morse homotopy

invariant of 3-manifolds (JAPANESE)

**Tadayuki Watanabe**(Shimane University)Higher-order generalization of Fukaya's Morse homotopy

invariant of 3-manifolds (JAPANESE)

[ Abstract ]

In his article published in 1996, K. Fukaya constructed

a 3-manifold invariant by using Morse homotopy theory. Roughly, his

invariant is defined by considering several Morse functions on a

3-manifold and counting with weights the ways that the theta-graph can

be immersed such that edges follow gradient lines. We generalize his

construction to 3-valent graphs with arbitrary number of loops for

integral homology 3-spheres. I will also discuss extension of our method

to 3-manifolds with positive first Betti numbers.

In his article published in 1996, K. Fukaya constructed

a 3-manifold invariant by using Morse homotopy theory. Roughly, his

invariant is defined by considering several Morse functions on a

3-manifold and counting with weights the ways that the theta-graph can

be immersed such that edges follow gradient lines. We generalize his

construction to 3-valent graphs with arbitrary number of loops for

integral homology 3-spheres. I will also discuss extension of our method

to 3-manifolds with positive first Betti numbers.

### 2013/06/18

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Left-orderable, non-L-space surgeries on knots (JAPANESE)

**Kimihiko Motegi**(Nihon University)Left-orderable, non-L-space surgeries on knots (JAPANESE)

[ Abstract ]

A Dehn surgery is said to be left-orderable

if the resulting manifold of the surgery has the left-orderable fundamental group,

and a Dehn surgery is called an L-space surgery

if the resulting manifold of the surgery is an L-space.

We will focus on left-orderable, non-L-space surgeries on knots in the 3-sphere.

Once we have a knot with left-orderable surgeries,

the ``periodic construction" enables us to provide infinitely many knots with

left-orderable, non-L-space surgeries.

We apply the construction to present infinitely many hyperbolic knots on each

of which every nontrivial surgery is a left-orderable, non-L-space surgery.

This is a joint work with Masakazu Teragaito.

A Dehn surgery is said to be left-orderable

if the resulting manifold of the surgery has the left-orderable fundamental group,

and a Dehn surgery is called an L-space surgery

if the resulting manifold of the surgery is an L-space.

We will focus on left-orderable, non-L-space surgeries on knots in the 3-sphere.

Once we have a knot with left-orderable surgeries,

the ``periodic construction" enables us to provide infinitely many knots with

left-orderable, non-L-space surgeries.

We apply the construction to present infinitely many hyperbolic knots on each

of which every nontrivial surgery is a left-orderable, non-L-space surgery.

This is a joint work with Masakazu Teragaito.

### 2013/06/11

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

On an analogue of Culler-Shalen theory for higher-dimensional

representations

(JAPANESE)

**Takahiro Kitayama**(The University of Tokyo)On an analogue of Culler-Shalen theory for higher-dimensional

representations

(JAPANESE)

[ Abstract ]

Culler and Shalen established a way to construct incompressible surfaces

in a 3-manifold from ideal points of the SL_2-character variety. We

present an analogous theory to construct certain kinds of branched

surfaces from limit points of the SL_n-character variety. Such a

branched surface induces a nontrivial presentation of the fundamental

group as a 2-dimensional complex of groups. This is a joint work with

Takashi Hara (Osaka University).

Culler and Shalen established a way to construct incompressible surfaces

in a 3-manifold from ideal points of the SL_2-character variety. We

present an analogous theory to construct certain kinds of branched

surfaces from limit points of the SL_n-character variety. Such a

branched surface induces a nontrivial presentation of the fundamental

group as a 2-dimensional complex of groups. This is a joint work with

Takashi Hara (Osaka University).

### 2013/06/04

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Low-dimensional linear representations of mapping class groups. (ENGLISH)

**Mustafa Korkmaz**(Middle East Technical University)Low-dimensional linear representations of mapping class groups. (ENGLISH)

[ Abstract ]

For a compact connected orientable surface, the mapping class group

of it is defined as the group of isotopy classes of orientation-preserving

self-diffeomorphisms of S which are identity on the boundary. The action

of the mapping class group on the first homology of the surface

gives rise to the classical 2g-dimensional symplectic representation.

The existence of a faithful linear representation of the mapping class

group is still unknown. In my talk, I will show the following three results;

there is no lower dimensional (complex) linear representation,

up to conjugation the symplectic representation is the unique nontrivial representation in dimension 2g, and there is no faithful linear representation

of the mapping class group in dimensions up to 3g-3. I will also discuss a few applications of these theorems, including some algebraic consequences.

For a compact connected orientable surface, the mapping class group

of it is defined as the group of isotopy classes of orientation-preserving

self-diffeomorphisms of S which are identity on the boundary. The action

of the mapping class group on the first homology of the surface

gives rise to the classical 2g-dimensional symplectic representation.

The existence of a faithful linear representation of the mapping class

group is still unknown. In my talk, I will show the following three results;

there is no lower dimensional (complex) linear representation,

up to conjugation the symplectic representation is the unique nontrivial representation in dimension 2g, and there is no faithful linear representation

of the mapping class group in dimensions up to 3g-3. I will also discuss a few applications of these theorems, including some algebraic consequences.

### 2013/05/21

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

A Heegaard Floer homology for bipartite spatial graphs and its

properties (ENGLISH)

**Yuanyuan Bao**(The University of Tokyo)A Heegaard Floer homology for bipartite spatial graphs and its

properties (ENGLISH)

[ Abstract ]

A spatial graph is a smooth embedding of a graph into a given

3-manifold. We can regard a link as a particular spatial graph.

So it is natural to ask whether it is possible to extend the idea

of link Floer homology to define a Heegaard Floer homology for

spatial graphs. In this talk, we discuss some ideas towards this

question. In particular, we define a Heegaard Floer homology for

bipartite spatial graphs and discuss some further observations

about this construction. We remark that Harvey and O’Donnol

have announced a combinatorial Floer homology for spatial graphs by

considering grid diagrams.

A spatial graph is a smooth embedding of a graph into a given

3-manifold. We can regard a link as a particular spatial graph.

So it is natural to ask whether it is possible to extend the idea

of link Floer homology to define a Heegaard Floer homology for

spatial graphs. In this talk, we discuss some ideas towards this

question. In particular, we define a Heegaard Floer homology for

bipartite spatial graphs and discuss some further observations

about this construction. We remark that Harvey and O’Donnol

have announced a combinatorial Floer homology for spatial graphs by

considering grid diagrams.

### 2013/05/14

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Vanishing cycles and homotopies of wrinkled fibrations (JAPANESE)

**Kenta Hayano**(Osaka University)Vanishing cycles and homotopies of wrinkled fibrations (JAPANESE)

[ Abstract ]

Wrinkled fibrations on closed 4-manifolds are stable

maps to closed surfaces with only indefinite singularities. Such

fibrations and variants of them have been studied for the past few years

to obtain new descriptions of 4-manifolds using mapping class groups.

Vanishing cycles of wrinkled fibrations play a key role in these studies.

In this talk, we will explain how homotopies of wrinkled fibrtions affect

their vanishing cycles. Part of the results in this talk is a joint work

with Stefan Behrens (Max Planck Institute for Mathematics).

Wrinkled fibrations on closed 4-manifolds are stable

maps to closed surfaces with only indefinite singularities. Such

fibrations and variants of them have been studied for the past few years

to obtain new descriptions of 4-manifolds using mapping class groups.

Vanishing cycles of wrinkled fibrations play a key role in these studies.

In this talk, we will explain how homotopies of wrinkled fibrtions affect

their vanishing cycles. Part of the results in this talk is a joint work

with Stefan Behrens (Max Planck Institute for Mathematics).

### 2013/05/07

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Homological intersection in braid group representation and dual

Garside structure (JAPANESE)

**Tetsuya Ito**(RIMS, Kyoto University)Homological intersection in braid group representation and dual

Garside structure (JAPANESE)

[ Abstract ]

One method to construct linear representations of braid groups is to use

an action of braid groups on certain homology of local system coefficient.

Many famous representations, such as Burau or Lawrence-Krammer-Bigelow

representations are constructed in such a way. We show that homological

intersections on such homology groups are closely related to the dual

Garside structure, a remarkable combinatorial structure of braid, and

prove that some representations detects the length of braids in a

surprisingly simple way.

This work is partially joint with Bert Wiest (Univ. Rennes1).

One method to construct linear representations of braid groups is to use

an action of braid groups on certain homology of local system coefficient.

Many famous representations, such as Burau or Lawrence-Krammer-Bigelow

representations are constructed in such a way. We show that homological

intersections on such homology groups are closely related to the dual

Garside structure, a remarkable combinatorial structure of braid, and

prove that some representations detects the length of braids in a

surprisingly simple way.

This work is partially joint with Bert Wiest (Univ. Rennes1).

### 2013/04/30

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Discrete vector fields and fundamental algebraic topology.

(ENGLISH)

**Francis Sergeraert**(L'Institut Fourier, Univ. de Grenoble)Discrete vector fields and fundamental algebraic topology.

(ENGLISH)

[ Abstract ]

Robin Forman invented the notion of Discrete Vector Field in 1997.

A recent common work with Ana Romero allowed us to discover the notion

of Eilenberg-Zilber discrete vector field. Giving the topologist a

totally new understanding of the fundamental tools of combinatorial

algebraic topology: Eilenberg-Zilber theorem, twisted Eilenberg-Zilber

theorem, Serre and Eilenberg-Moore spectral sequences,

Eilenberg-MacLane correspondence between topological and algebraic

classifying spaces. Gives also new efficient algorithms for Algebraic

Topology, considerably improving our computer program Kenzo, devoted

to Constructive Algebraic Topology. The talk is devoted to an

introduction to discrete vector fields, the very simple definition of

the Eilenberg-Zilber vector field, and how it can be used in various

contexts.

Robin Forman invented the notion of Discrete Vector Field in 1997.

A recent common work with Ana Romero allowed us to discover the notion

of Eilenberg-Zilber discrete vector field. Giving the topologist a

totally new understanding of the fundamental tools of combinatorial

algebraic topology: Eilenberg-Zilber theorem, twisted Eilenberg-Zilber

theorem, Serre and Eilenberg-Moore spectral sequences,

Eilenberg-MacLane correspondence between topological and algebraic

classifying spaces. Gives also new efficient algorithms for Algebraic

Topology, considerably improving our computer program Kenzo, devoted

to Constructive Algebraic Topology. The talk is devoted to an

introduction to discrete vector fields, the very simple definition of

the Eilenberg-Zilber vector field, and how it can be used in various

contexts.