## Tuesday Seminar on Topology

Seminar information archive ～09/17｜Next seminar｜Future seminars 09/18～

Date, time & place | Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
---|---|

Organizer(s) | KOHNO Toshitake, KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya |

Remarks | Tea: 16:30 - 17:00 Common Room |

**Seminar information archive**

### 2009/12/01

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

Non-Abelian Novikov homology

**Andrei Pajitnov**(Univ. de Nantes)Non-Abelian Novikov homology

[ Abstract ]

Classical construction of S.P. Novikov

associates to each circle-valued Morse map

a chain complex defined over a ring

of Laurent power series in one variable.

In this survey talk we shall explain several

results related to the construction and

properties of non-Abelian generalizations of the

Novikov complex.

Classical construction of S.P. Novikov

associates to each circle-valued Morse map

a chain complex defined over a ring

of Laurent power series in one variable.

In this survey talk we shall explain several

results related to the construction and

properties of non-Abelian generalizations of the

Novikov complex.

### 2009/11/24

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

A topological approach to left orderable groups

**Adam Clay**(University of British Columbia)A topological approach to left orderable groups

[ Abstract ]

A group G is said to be left orderable if there is a strict

total ordering of its elements such that gin G. Left orderable groups have been useful in solving many problems in topology, and now we find that topology is returning the favour: the set of all left orderings of a group is denoted by LO(G), and it admits a natural topology, under which LO(G) becomes a compact topological

space. In general, the structure of the space LO(G) is not well understood, although there are surprising results in a few special cases.

For example, the space of left orderings of the braid group B_n for n>2

contains isolated points (yet it is uncountable), while the space of left

orderings of the fundamental group of the Klein bottle is finite.

Twice in recent years, the space of left orderings has been used very

successfully to solve difficult open problems from the field of left

orderable groups, even though the connection between the topology of LO(G) and the algebraic properties of G was still unclear. I will explain the

newest understanding of this connection, and highlight some potential

applications of further advances.

A group G is said to be left orderable if there is a strict

total ordering of its elements such that g

space. In general, the structure of the space LO(G) is not well understood, although there are surprising results in a few special cases.

For example, the space of left orderings of the braid group B_n for n>2

contains isolated points (yet it is uncountable), while the space of left

orderings of the fundamental group of the Klein bottle is finite.

Twice in recent years, the space of left orderings has been used very

successfully to solve difficult open problems from the field of left

orderable groups, even though the connection between the topology of LO(G) and the algebraic properties of G was still unclear. I will explain the

newest understanding of this connection, and highlight some potential

applications of further advances.

### 2009/11/17

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

On the $SO(N)$ and $Sp(N)$ free energy of a closed oriented 3-manifold

**高田 敏恵**(新潟大学)On the $SO(N)$ and $Sp(N)$ free energy of a closed oriented 3-manifold

[ Abstract ]

We give an explicit formula of the $SO(N)$ and $Sp(N)$ free energy

of a lens space and show that the genus $g$ terms of it are analytic

in a neighborhood at zero, where we can choose the neighborhood

independently of $g$.

Moreover, it is proved that for any closed oriented 3-manifold $M$

and any $g$, the genus $g$ terms of $SO(N)$ and $Sp(N)$ free energy

of $M$ coincide up to sign.

We give an explicit formula of the $SO(N)$ and $Sp(N)$ free energy

of a lens space and show that the genus $g$ terms of it are analytic

in a neighborhood at zero, where we can choose the neighborhood

independently of $g$.

Moreover, it is proved that for any closed oriented 3-manifold $M$

and any $g$, the genus $g$ terms of $SO(N)$ and $Sp(N)$ free energy

of $M$ coincide up to sign.

### 2009/11/10

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

Resurgent analysis of the Witten Laplacian in one dimension

**Alexander Getmanenko**(IPMU)Resurgent analysis of the Witten Laplacian in one dimension

[ Abstract ]

I will recall Witten's approach to the Morse theory through properties of a certain differential operator. Then I will introduce resurgent analysis -- an asymptotic method used, in particular, for studying quantum-mechanical tunneling. In conclusion I will discuss how the methods of resurgent analysis can help us "see" pseudoholomorphic discs in the eigenfunctions of the Witten Laplacian.

I will recall Witten's approach to the Morse theory through properties of a certain differential operator. Then I will introduce resurgent analysis -- an asymptotic method used, in particular, for studying quantum-mechanical tunneling. In conclusion I will discuss how the methods of resurgent analysis can help us "see" pseudoholomorphic discs in the eigenfunctions of the Witten Laplacian.

### 2009/10/27

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

A new appearance of the Morita-Penner cocycle

**Alex Bene**(IPMU)A new appearance of the Morita-Penner cocycle

[ Abstract ]

In this talk, I will recall the Morita-Penner cocycle on the dual fatgraph complex for a surface with one boundary component. This cocycle, when restricted to paths representing elements of the mapping class group, represents the extended first Johnson homomorphism \\tau_1, thus can be viewed as a (in some specific sense canonical) "groupoid extension" of \\tau_1. There are now several different contexts in which this cocycle can be constructed, and in this talk I will briefly review several of them, including one discovered in the context of finite type invariants of homology cylinders in joint work with J.E. Andersen, J-B. Meilhan, and R.C. Penner.

In this talk, I will recall the Morita-Penner cocycle on the dual fatgraph complex for a surface with one boundary component. This cocycle, when restricted to paths representing elements of the mapping class group, represents the extended first Johnson homomorphism \\tau_1, thus can be viewed as a (in some specific sense canonical) "groupoid extension" of \\tau_1. There are now several different contexts in which this cocycle can be constructed, and in this talk I will briefly review several of them, including one discovered in the context of finite type invariants of homology cylinders in joint work with J.E. Andersen, J-B. Meilhan, and R.C. Penner.

### 2009/10/20

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

Torus fibrations and localization of index

**吉田 尚彦**(明治大学大学院理工学研究科)Torus fibrations and localization of index

[ Abstract ]

I will describe a localization of index of a Dirac type operator.

We make use of a structure of torus fibration, but the mechanism

of the localization does not rely on any group action. In the case of

Lagrangian fibration, we show that the index is described as a sum of

the contributions from Bohr-Sommerfeld fibers and singular fibers.

To show the localization we introduce a deformation of a Dirac type

operator for a manifold equipped with a fiber bundle structure which

satisfies a kind of acyclic condition. The deformation allows an

interpretation as an adiabatic limit or an infinite dimensional analogue

of Witten deformation.

Joint work with Hajime Fujita and Mikio Furuta.

I will describe a localization of index of a Dirac type operator.

We make use of a structure of torus fibration, but the mechanism

of the localization does not rely on any group action. In the case of

Lagrangian fibration, we show that the index is described as a sum of

the contributions from Bohr-Sommerfeld fibers and singular fibers.

To show the localization we introduce a deformation of a Dirac type

operator for a manifold equipped with a fiber bundle structure which

satisfies a kind of acyclic condition. The deformation allows an

interpretation as an adiabatic limit or an infinite dimensional analogue

of Witten deformation.

Joint work with Hajime Fujita and Mikio Furuta.

### 2009/10/13

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

Instanton Floer homology for lens spaces

**笹平 裕史**(東京大学大学院数理科学研究科)Instanton Floer homology for lens spaces

[ Abstract ]

Let Y be an oriented closed 3-manifold and P be an SU(2)-bundle on Y. Under a certain condition, instanton Floer homology for Y can be defined as the Morse homology of the Chern-Simons functional. The condition is that all flat connections on P are irreducible. When there is a reducible flat connection on P, instanton Floer homology is not defined in general.

Since the fundamental group of a lens sapce is commutative, all flat connections on the lens space are reducible. In this talk I will introduce instanton Floer homology for lens spaces. I also show calculations for some lens spaces.

Let Y be an oriented closed 3-manifold and P be an SU(2)-bundle on Y. Under a certain condition, instanton Floer homology for Y can be defined as the Morse homology of the Chern-Simons functional. The condition is that all flat connections on P are irreducible. When there is a reducible flat connection on P, instanton Floer homology is not defined in general.

Since the fundamental group of a lens sapce is commutative, all flat connections on the lens space are reducible. In this talk I will introduce instanton Floer homology for lens spaces. I also show calculations for some lens spaces.

### 2009/09/29

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

Symbol of the Conway polynomial and Drinfeld associator

**Sergei Duzhin**(Steklov Mathematical Institute, Petersburg Division)Symbol of the Conway polynomial and Drinfeld associator

[ Abstract ]

The Magnus expansion is a universal finite type invariant of pure braids

with values in the space of horizontal chord diagrams. The Conway polynomial

composed with the short circuit map from braids to knots gives rise to a

series of finite type invariants of pure braids and thus factors through

the Magnus map. We describe explicitly the resulting mapping from horizontal

chord diagrams on 3 strands to univariante polynomials and evaluate it on

the Drinfeld associator obtaining a beautiful generating function whose

coefficients are integer combinations of multple zeta values.

The Magnus expansion is a universal finite type invariant of pure braids

with values in the space of horizontal chord diagrams. The Conway polynomial

composed with the short circuit map from braids to knots gives rise to a

series of finite type invariants of pure braids and thus factors through

the Magnus map. We describe explicitly the resulting mapping from horizontal

chord diagrams on 3 strands to univariante polynomials and evaluate it on

the Drinfeld associator obtaining a beautiful generating function whose

coefficients are integer combinations of multple zeta values.

### 2009/07/14

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

The Cannon-Thurston maps and the canonical decompositions

of punctured-torus bundles over the circle.

**作間 誠**(広島大学)The Cannon-Thurston maps and the canonical decompositions

of punctured-torus bundles over the circle.

[ Abstract ]

To each once-punctured-torus bundle over the circle

with pseudo-Anosov monodromy, there are associated two tessellations of the complex plane:

one is the triangulation of a horosphere induced by the canonical decomposition into ideal

tetrahedra, and the other is a fractal tessellation

given by the Cannon-Thurston map of the fiber group.

In this talk, I will explain the relation between these two tessellations

(joint work with Warren Dicks).

I will also explain the relation of the fractal tessellation and

the "circle chains" of double cusp groups converging to the fiber group

(joint work with Caroline Series).

If time permits, I would like to discuss possible generalization of these results

to higher-genus punctured surface bundles.

To each once-punctured-torus bundle over the circle

with pseudo-Anosov monodromy, there are associated two tessellations of the complex plane:

one is the triangulation of a horosphere induced by the canonical decomposition into ideal

tetrahedra, and the other is a fractal tessellation

given by the Cannon-Thurston map of the fiber group.

In this talk, I will explain the relation between these two tessellations

(joint work with Warren Dicks).

I will also explain the relation of the fractal tessellation and

the "circle chains" of double cusp groups converging to the fiber group

(joint work with Caroline Series).

If time permits, I would like to discuss possible generalization of these results

to higher-genus punctured surface bundles.

### 2009/06/30

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

Torsion volume forms and twisted Alexander functions on

character varieties of knots

**北山 貴裕**(東京大学大学院数理科学研究科)Torsion volume forms and twisted Alexander functions on

character varieties of knots

[ Abstract ]

Using non-acyclic Reidemeister torsion, we can canonically

construct a complex volume form on each component of the

lowest dimension of the $SL_2(\\mathbb{C})$-character

variety of a link group.

This volume form enjoys a certain compatibility with the

following natural transformations on the variety.

Two of them are involutions which come from the algebraic

structure of $SL_2(\\mathbb{C})$ and the other is the

action by the outer automorphism group of the link group.

Moreover, in the case of knots these results deduce a kind

of symmetry of the $SU_2$-twisted Alexander functions

which are globally described via the volume form.

Using non-acyclic Reidemeister torsion, we can canonically

construct a complex volume form on each component of the

lowest dimension of the $SL_2(\\mathbb{C})$-character

variety of a link group.

This volume form enjoys a certain compatibility with the

following natural transformations on the variety.

Two of them are involutions which come from the algebraic

structure of $SL_2(\\mathbb{C})$ and the other is the

action by the outer automorphism group of the link group.

Moreover, in the case of knots these results deduce a kind

of symmetry of the $SU_2$-twisted Alexander functions

which are globally described via the volume form.

### 2009/06/23

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

The Meyer functions for projective varieties and their applications

**久野 雄介**(東京大学大学院数理科学研究科)The Meyer functions for projective varieties and their applications

[ Abstract ]

Meyer function is a kind of secondary invariant related to the signature

of surface bundles over surfaces. In this talk I will show there exist uniquely the Meyer function

for each smooth projective variety.

Our function is a class function on the fundamental group of some open algebraic variety.

I will also talk about its application to local signature for fibered 4-manifolds

Meyer function is a kind of secondary invariant related to the signature

of surface bundles over surfaces. In this talk I will show there exist uniquely the Meyer function

for each smooth projective variety.

Our function is a class function on the fundamental group of some open algebraic variety.

I will also talk about its application to local signature for fibered 4-manifolds

### 2009/06/16

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

The abelianization of the level 2 mapping class group

**佐藤 正寿**(東京大学大学院数理科学研究科)The abelianization of the level 2 mapping class group

[ Abstract ]

The level d mapping class group is a finite index subgroup of the mapping class group of an orientable closed surface. For d greater than or equal to 3, the abelianization of this group is equal to the first homology group of the moduli space of nonsingular curves with level d structure.

In this talk, we determine the abelianization of the level d mapping class group for d=2 and odd d. For even d greater than 2, we also determine it up to a cyclic group of order 2.

The level d mapping class group is a finite index subgroup of the mapping class group of an orientable closed surface. For d greater than or equal to 3, the abelianization of this group is equal to the first homology group of the moduli space of nonsingular curves with level d structure.

In this talk, we determine the abelianization of the level d mapping class group for d=2 and odd d. For even d greater than 2, we also determine it up to a cyclic group of order 2.

### 2009/06/09

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

A finite-dimensional construction of the Chern character for

twisted K-theory

**五味 清紀**(京都大学大学院理学研究科)A finite-dimensional construction of the Chern character for

twisted K-theory

[ Abstract ]

Twisted K-theory is a variant of topological K-theory, and

is attracting much interest due to applications to physics recently.

Usually, twisted K-theory is formulated infinite-dimensionally, and

hence known constructions of its Chern character are more or less

abstract. The aim of my talk is to explain a purely finite-dimensional

construction of the Chern character for twisted K-theory, which allows

us to compute examples concretely. The construction is based on

twisted version of Furuta's generalized vector bundle, and Quillen's

superconnection.

This is a joint work with Yuji Terashima.

Twisted K-theory is a variant of topological K-theory, and

is attracting much interest due to applications to physics recently.

Usually, twisted K-theory is formulated infinite-dimensionally, and

hence known constructions of its Chern character are more or less

abstract. The aim of my talk is to explain a purely finite-dimensional

construction of the Chern character for twisted K-theory, which allows

us to compute examples concretely. The construction is based on

twisted version of Furuta's generalized vector bundle, and Quillen's

superconnection.

This is a joint work with Yuji Terashima.

### 2009/06/02

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

Graph homology: Koszul duality = Verdier duality

**Alexander Voronov**(University of Minnesota)Graph homology: Koszul duality = Verdier duality

[ Abstract ]

Graph cohomology appears in computation of the cohomology of the moduli space of Riemann surfaces and the outer automorphism group of a free group. In the former case, it is graph cohomology of the commutative and Lie types, in the latter it is ribbon graph cohomology, that is to say, graph cohomology of the associative type. The presence of these three basic types of algebraic structures hints at a relation between Koszul duality for operads and Poincare-Lefschetz duality for manifolds. I will show how the more general Verdier duality for certain sheaves on the moduli spaces of graphs associated to Koszul operads corresponds to Koszul duality of operads. This is a joint work with Andrey Lazarev.

Graph cohomology appears in computation of the cohomology of the moduli space of Riemann surfaces and the outer automorphism group of a free group. In the former case, it is graph cohomology of the commutative and Lie types, in the latter it is ribbon graph cohomology, that is to say, graph cohomology of the associative type. The presence of these three basic types of algebraic structures hints at a relation between Koszul duality for operads and Poincare-Lefschetz duality for manifolds. I will show how the more general Verdier duality for certain sheaves on the moduli spaces of graphs associated to Koszul operads corresponds to Koszul duality of operads. This is a joint work with Andrey Lazarev.

### 2009/05/26

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

Configuration space integrals and the cohomology of the space of long embeddings

**境 圭一**(東京大学大学院数理科学研究科)Configuration space integrals and the cohomology of the space of long embeddings

[ Abstract ]

It is known that some non-trivial cohomology classes, such as finite type invariants for (long) 1-knots (Bott-Taubes, Kohno, ...) and invariants for codimension two, odd dimensional long embeddings (Bott, Cattaneo-Rossi, Watanabe) are given as configuration space integrals associated with trivalent graphs.

In this talk, I will describe more cohomology classes by means of configuration space integral, in particular those arising from non-trivalent graphs and a new formulation of the Haefliger invariant for long 3-embeddings in 6-space, in relation to Budney's little balls operad action and Roseman-Takase's deform-spinning.

This is in part a joint work with Tadayuki Watanabe.

It is known that some non-trivial cohomology classes, such as finite type invariants for (long) 1-knots (Bott-Taubes, Kohno, ...) and invariants for codimension two, odd dimensional long embeddings (Bott, Cattaneo-Rossi, Watanabe) are given as configuration space integrals associated with trivalent graphs.

In this talk, I will describe more cohomology classes by means of configuration space integral, in particular those arising from non-trivalent graphs and a new formulation of the Haefliger invariant for long 3-embeddings in 6-space, in relation to Budney's little balls operad action and Roseman-Takase's deform-spinning.

This is in part a joint work with Tadayuki Watanabe.

### 2009/05/19

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

Geometric quantization of integrable systems

**Mark Hamilton**(東京大学大学院数理科学研究科, JSPS)Geometric quantization of integrable systems

[ Abstract ]

The theory of geometric quantization is one way of producing a "quantum system" from a "classical system," and has been studied a great deal over the past several decades. It also has surprising ties to representation theory. However, despite this, there still does not exist a satisfactory theory of quantization for systems with singularities.

Geometric quantization requires the choice of a polarization; when using a real polarization to quantize a regular enough manifold, a result of Sniatycki says that the quantization can be found by counting certain objects, called Bohr-Sommerfeld fibres. However, there are many types of systems to which this result does not apply. One such type is the class of completely integrable systems, which are examples coming from mechanics that have many nice properties, but which are nontheless too singular for Sniatycki's theorem to apply.

In this talk we will explore one approach to the quantization of integrable systems, and show a Sniatycki-type relationship to Bohr-Sommerfeld fibres. However, some surprising features appear, including infinite-dimensional contributions and strong dependence on the polarization.

This is joint work with Eva Miranda.

The theory of geometric quantization is one way of producing a "quantum system" from a "classical system," and has been studied a great deal over the past several decades. It also has surprising ties to representation theory. However, despite this, there still does not exist a satisfactory theory of quantization for systems with singularities.

Geometric quantization requires the choice of a polarization; when using a real polarization to quantize a regular enough manifold, a result of Sniatycki says that the quantization can be found by counting certain objects, called Bohr-Sommerfeld fibres. However, there are many types of systems to which this result does not apply. One such type is the class of completely integrable systems, which are examples coming from mechanics that have many nice properties, but which are nontheless too singular for Sniatycki's theorem to apply.

In this talk we will explore one approach to the quantization of integrable systems, and show a Sniatycki-type relationship to Bohr-Sommerfeld fibres. However, some surprising features appear, including infinite-dimensional contributions and strong dependence on the polarization.

This is joint work with Eva Miranda.

### 2009/05/12

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

Discrete subgroups of the group of circle diffeomorphisms

**松田 能文**(東京大学大学院数理科学研究科)Discrete subgroups of the group of circle diffeomorphisms

[ Abstract ]

Typical examples of discrete subgroups of the group of circle diffeomorphisms

are Fuchsian groups.

In this talk, we construct discrete subgroups of the group of

orientation-preserving

real analytic cirlcle diffeomorphisms

which are not topologically conjugate to finite coverings of Fuchsian groups.

Typical examples of discrete subgroups of the group of circle diffeomorphisms

are Fuchsian groups.

In this talk, we construct discrete subgroups of the group of

orientation-preserving

real analytic cirlcle diffeomorphisms

which are not topologically conjugate to finite coverings of Fuchsian groups.

### 2009/04/28

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

The ambient metric in conformal geometry

**平地 健吾**(東京大学大学院数理科学研究科)The ambient metric in conformal geometry

[ Abstract ]

In 1985, Charles Fefferman and Robin Graham gave a method for realizing a conformal manifold of dimension n as a submanifold of a Ricci-flat Lorentz metric on a manifold of dimension n+2, which is now called the ambient space. Using this correspondence, one can construct many examples of conformal invariants and conformally invariant operators. However, if n is even, their construction of the ambient space is obstructed at the jet of order n/2 and thereby the application of the ambient space was limited. In this talk, I'll recall basic ideas of the ambient space and then explain how to avoid the difficulty and go beyond the obstruction. This is a joint work with Robin Graham.

In 1985, Charles Fefferman and Robin Graham gave a method for realizing a conformal manifold of dimension n as a submanifold of a Ricci-flat Lorentz metric on a manifold of dimension n+2, which is now called the ambient space. Using this correspondence, one can construct many examples of conformal invariants and conformally invariant operators. However, if n is even, their construction of the ambient space is obstructed at the jet of order n/2 and thereby the application of the ambient space was limited. In this talk, I'll recall basic ideas of the ambient space and then explain how to avoid the difficulty and go beyond the obstruction. This is a joint work with Robin Graham.

### 2009/04/21

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

Some algebraic aspects of KZ systems

**Ivan Marin**(Univ. Paris VII)Some algebraic aspects of KZ systems

[ Abstract ]

Knizhnik-Zamolodchikov (KZ) systems enables one

to construct representations of (generalized)

braid groups. While this geometric construction is

now very well understood, it still brings to

attention, or helps constructing, new algebraic objects.

In this talk, we will present some of them, including an

infinitesimal version of Iwahori-Hecke algebras and a

generalization of the Krammer representations of the usual

braid groups.

Knizhnik-Zamolodchikov (KZ) systems enables one

to construct representations of (generalized)

braid groups. While this geometric construction is

now very well understood, it still brings to

attention, or helps constructing, new algebraic objects.

In this talk, we will present some of them, including an

infinitesimal version of Iwahori-Hecke algebras and a

generalization of the Krammer representations of the usual

braid groups.

### 2009/03/05

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

Extending surface automorphisms over 4-space

**Shicheng Wang**(Peking University)Extending surface automorphisms over 4-space

[ Abstract ]

Let $e: M^p\\to R^{p+2}$ be a co-dimensional 2 smooth embedding

from a closed orientable manifold to the Euclidean space and $E_e$ be the subgroup of ${\\cal M}_M$, the mapping class group

of $M$, whose elements extend over $R^{p+2}$ as self-diffeomorphisms. Then there is a spin structure

on $M$ derived from the embedding which is preserved by each $\\tau \\in E_e$.

Some applications: (1) the index $[{\\cal M}_{F_g}:E_e]\\geq 2^{2g-1}+2^{g-1}$ for any embedding $e:F_g\\to R^4$, where $F_g$

is the surface of genus $g$. (2) $[{\\cal M}_{T^p}:E_e]\\geq 2^p-1$ for any unknotted embedding

$e:T^p\\to R^{p+2}$, where $T^p$ is the $p$-dimensional torus. Those two lower bounds are known to be sharp.

This is a joint work of Ding-Liu-Wang-Yao.

Let $e: M^p\\to R^{p+2}$ be a co-dimensional 2 smooth embedding

from a closed orientable manifold to the Euclidean space and $E_e$ be the subgroup of ${\\cal M}_M$, the mapping class group

of $M$, whose elements extend over $R^{p+2}$ as self-diffeomorphisms. Then there is a spin structure

on $M$ derived from the embedding which is preserved by each $\\tau \\in E_e$.

Some applications: (1) the index $[{\\cal M}_{F_g}:E_e]\\geq 2^{2g-1}+2^{g-1}$ for any embedding $e:F_g\\to R^4$, where $F_g$

is the surface of genus $g$. (2) $[{\\cal M}_{T^p}:E_e]\\geq 2^p-1$ for any unknotted embedding

$e:T^p\\to R^{p+2}$, where $T^p$ is the $p$-dimensional torus. Those two lower bounds are known to be sharp.

This is a joint work of Ding-Liu-Wang-Yao.

### 2009/01/27

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

Lagrangian Floer homology and quasi homomorphism

from the group of Hamiltonian diffeomorphism

**深谷 賢治**(京都大学大学院理学研究科)Lagrangian Floer homology and quasi homomorphism

from the group of Hamiltonian diffeomorphism

[ Abstract ]

Entov-Polterovich constructed quasi homomorphism

from the group of Hamiltonian diffeomorphisms using

spectral invariant due to Oh etc.

In this talk I will explain a way to study this

quasi homomorphism by using Lagrangian Floer homology.

I will also explain its generalization to use quantum

cohomology with bulk deformation.

When applied to the case of toric manifold, it

gives an example where (infinitely) many quasi homomorphism

exists.

(Joint work with Oh-Ohta-Ono).

Entov-Polterovich constructed quasi homomorphism

from the group of Hamiltonian diffeomorphisms using

spectral invariant due to Oh etc.

In this talk I will explain a way to study this

quasi homomorphism by using Lagrangian Floer homology.

I will also explain its generalization to use quantum

cohomology with bulk deformation.

When applied to the case of toric manifold, it

gives an example where (infinitely) many quasi homomorphism

exists.

(Joint work with Oh-Ohta-Ono).

### 2009/01/20

16:30-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Five dimensional $K$-contact manifolds of rank 2

**野澤 啓**(東京大学大学院数理科学研究科)Five dimensional $K$-contact manifolds of rank 2

[ Abstract ]

A $K$-contact manifold is an odd dimensional manifold $M$ with a contact form $\\alpha$ whose Reeb flow preserves a Riemannian metric on $M$. For examples, the underlying manifold with the underlying contact form of a Sasakian manifold is $K$-contact. In this talk, we will state our results on classification up to surgeries, the existence of compatible Sasakian metrics and a sufficient condition to be toric for closed $5$-dimensional $K$-contact manifolds with a $T^2$ action given by the closure of the Reeb flow, which are obtained by the application of Morse theory to the contact moment map for the $T^2$ action.

A $K$-contact manifold is an odd dimensional manifold $M$ with a contact form $\\alpha$ whose Reeb flow preserves a Riemannian metric on $M$. For examples, the underlying manifold with the underlying contact form of a Sasakian manifold is $K$-contact. In this talk, we will state our results on classification up to surgeries, the existence of compatible Sasakian metrics and a sufficient condition to be toric for closed $5$-dimensional $K$-contact manifolds with a $T^2$ action given by the closure of the Reeb flow, which are obtained by the application of Morse theory to the contact moment map for the $T^2$ action.

### 2009/01/20

17:30-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Surface links which are coverings of a trivial torus knot (JAPANESE)

**中村 伊南沙**(東京大学大学院数理科学研究科)Surface links which are coverings of a trivial torus knot (JAPANESE)

[ Abstract ]

We consider surface links which are in the form of coverings of a

trivial torus knot, which we will call torus-covering-links.

By definition, torus-covering-links include

spun $T^2$-knots, turned spun $T^2$-knots, and symmetry-spun tori.

We see some properties of torus-covering-links.

We consider surface links which are in the form of coverings of a

trivial torus knot, which we will call torus-covering-links.

By definition, torus-covering-links include

spun $T^2$-knots, turned spun $T^2$-knots, and symmetry-spun tori.

We see some properties of torus-covering-links.

### 2009/01/13

16:30-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Compactification of the homeomorphism group of a graph

**山下 温**(東京大学大学院数理科学研究科)Compactification of the homeomorphism group of a graph

[ Abstract ]

Topological properties of homeomorphism groups, especially of finite-dimensional manifolds,

have been of interest in the area of infinite-dimensional manifold topology.

For a locally finite graph $\\Gamma$ with countably many components,

the homeomorphism group $\\mathcal{H}(\\Gamma)$

and its identity component $\\mathcal{H}_+(\\Gamma)$ are topological groups

with respect to the compact-open topology. I will define natural compactifications

$\\overline{\\mathcal{H}}(\\Gamma)$ and

$\\overline{\\mathcal{H}}_+(\\Gamma)$ of these groups and describe the

topological type of the pair $(\\overline{\\mathcal{H}}_+(\\Gamma), \\mathcal{H}_+(\\Gamma))$

using the data of $\\Gamma$. I will also discuss the topological structure of

$\\overline{\\mathcal{H}}(\\Gamma)$ where $\\Gamma$ is the circle.

Topological properties of homeomorphism groups, especially of finite-dimensional manifolds,

have been of interest in the area of infinite-dimensional manifold topology.

For a locally finite graph $\\Gamma$ with countably many components,

the homeomorphism group $\\mathcal{H}(\\Gamma)$

and its identity component $\\mathcal{H}_+(\\Gamma)$ are topological groups

with respect to the compact-open topology. I will define natural compactifications

$\\overline{\\mathcal{H}}(\\Gamma)$ and

$\\overline{\\mathcal{H}}_+(\\Gamma)$ of these groups and describe the

topological type of the pair $(\\overline{\\mathcal{H}}_+(\\Gamma), \\mathcal{H}_+(\\Gamma))$

using the data of $\\Gamma$. I will also discuss the topological structure of

$\\overline{\\mathcal{H}}(\\Gamma)$ where $\\Gamma$ is the circle.

### 2008/12/09

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

Tits alternative in $Diff^1(S^1)$

**Bertrand Deroin**(CNRS, Orsay, Universit\'e Paris-Sud 11)Tits alternative in $Diff^1(S^1)$

[ Abstract ]

The following form of Tits alternative for subgroups of

homeomorphisms of the circle has been proved by Margulis: or the group

preserve a probability measure on the circle, or it contains a free

subgroup on two generators. We will prove that if the group acts by diffeomorphisms of

class $C^1$ and does not preserve a probability measure on the circle, then

in fact it contains a subgroup topologically conjugated to a Schottky group.

This is a joint work with V. Kleptsyn and A. Navas.

The following form of Tits alternative for subgroups of

homeomorphisms of the circle has been proved by Margulis: or the group

preserve a probability measure on the circle, or it contains a free

subgroup on two generators. We will prove that if the group acts by diffeomorphisms of

class $C^1$ and does not preserve a probability measure on the circle, then

in fact it contains a subgroup topologically conjugated to a Schottky group.

This is a joint work with V. Kleptsyn and A. Navas.