## Tuesday Seminar on Topology

Seminar information archive ～02/06｜Next seminar｜Future seminars 02/07～

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**

### 2007/01/23

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

The twistor correspondence for self-dual Zollfrei metrics

----their singularities and reduction

On the homology group of $Out(F_n)$

**中田 文憲**(東京大学大学院数理科学研究科) 16:30-17:30The twistor correspondence for self-dual Zollfrei metrics

----their singularities and reduction

[ Abstract ]

C. LeBrun and L. J. Mason investigated a twistor-type correspondence

between indefinite conformal structures of signature (2,2) with some properties

and totally real embeddings from RP^3 to CP^3.

In this talk, two themes following LeBrun and Mason are explained.

First we consider an additional structure:

the conformal structure is equipped with a null surface foliation

which has some singularity.

We establish a global twistor correspondence for such structures,

and we show that a low dimensional correspondence

between some quotient spaces is induced from this twistor correspondence.

Next we formulate a certain singularity for the conformal structures.

We generalize the formulation of LeBrun and Mason's theorem

admitting the singularity, and we show explicit examples.

C. LeBrun and L. J. Mason investigated a twistor-type correspondence

between indefinite conformal structures of signature (2,2) with some properties

and totally real embeddings from RP^3 to CP^3.

In this talk, two themes following LeBrun and Mason are explained.

First we consider an additional structure:

the conformal structure is equipped with a null surface foliation

which has some singularity.

We establish a global twistor correspondence for such structures,

and we show that a low dimensional correspondence

between some quotient spaces is induced from this twistor correspondence.

Next we formulate a certain singularity for the conformal structures.

We generalize the formulation of LeBrun and Mason's theorem

admitting the singularity, and we show explicit examples.

**大橋 了**(東京大学大学院数理科学研究科) 17:30-18:30On the homology group of $Out(F_n)$

[ Abstract ]

Suppose $F_n$ is the free group of rank $n$,

$Out(F_n) = Aut(F_n)/Inn(F_n)$ the outer automorphism group of $F_n$.

We compute $H_*(Out(F_n);\\mathbb{Q})$ for $n\\leq 6$ and conclude

that non-trivial classes in this range are generated

by Morita classes $\\mu_i \\in H_{4i}(Out(F_{2i+2});\\mathbb{Q})$.

Also we compute odd primary part of $H^*(Out(F_4);\\mathbb{Z})$.

Suppose $F_n$ is the free group of rank $n$,

$Out(F_n) = Aut(F_n)/Inn(F_n)$ the outer automorphism group of $F_n$.

We compute $H_*(Out(F_n);\\mathbb{Q})$ for $n\\leq 6$ and conclude

that non-trivial classes in this range are generated

by Morita classes $\\mu_i \\in H_{4i}(Out(F_{2i+2});\\mathbb{Q})$.

Also we compute odd primary part of $H^*(Out(F_4);\\mathbb{Z})$.

### 2007/01/16

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

An $SO(3)$-version of $2$-torsion instanton invariants

On the non-acyclic Reidemeister torsion for knots

**笹平 裕史**(東京大学大学院数理科学研究科) 16:30-17:30An $SO(3)$-version of $2$-torsion instanton invariants

[ Abstract ]

We construct invariants for simply connected, non-spin $4$-manifolds using torsion cohomology classes of moduli spaces of ASD connections on $SO(3)$-bundles. The invariants are $SO(3)$-version of Fintushel-Stern's $2$-torsion instanton invariants. We show that this $SO(3)$-torsion invariant of $2CP^2 \\# -CP^2$ is non-trivial, while it is known that any invariants of $2CP^2 \\# - CP^2$ coming from the Seiberg-Witten theory are trivial

since $2CP^2 \\# -CP^2$ has a positive scalar curvature metric.

We construct invariants for simply connected, non-spin $4$-manifolds using torsion cohomology classes of moduli spaces of ASD connections on $SO(3)$-bundles. The invariants are $SO(3)$-version of Fintushel-Stern's $2$-torsion instanton invariants. We show that this $SO(3)$-torsion invariant of $2CP^2 \\# -CP^2$ is non-trivial, while it is known that any invariants of $2CP^2 \\# - CP^2$ coming from the Seiberg-Witten theory are trivial

since $2CP^2 \\# -CP^2$ has a positive scalar curvature metric.

**山口 祥司**(東京大学大学院数理科学研究科) 17:30-18:30On the non-acyclic Reidemeister torsion for knots

[ Abstract ]

The Reidemeister torsion is an invariant of a CW-complex and a representation of its fundamental group. We consider the Reidemeister torsion for a knot exterior in a homology three sphere and a representation given by the composition of an SL(2, C)- (or SU(2)-) representation of the knot group and the adjoint action to the Lie algebra.

We will see that this invariant is expressed by the differential coefficient of the twisted Alexander invariant of the knot and investigate some properties of the invariant by using this relation.

The Reidemeister torsion is an invariant of a CW-complex and a representation of its fundamental group. We consider the Reidemeister torsion for a knot exterior in a homology three sphere and a representation given by the composition of an SL(2, C)- (or SU(2)-) representation of the knot group and the adjoint action to the Lie algebra.

We will see that this invariant is expressed by the differential coefficient of the twisted Alexander invariant of the knot and investigate some properties of the invariant by using this relation.

### 2006/12/19

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

Poisson structures on the homology of the spaces of knots

On projections of pseudo-ribbon sphere-links

**境 圭一**(東京大学大学院数理科学研究科) 16:30-17:30Poisson structures on the homology of the spaces of knots

[ Abstract ]

We study the homological properties of the space $K$ of (framed) long knots in $\\R^n$, $n>3$, in particular its Poisson algebra structures.

We had known two kinds of Poisson structures, both of which are based on the action of little disks operad. One definition is via the action on the space $K$. Another comes from the action of chains of little disks on the Hochschild complex of an operad, which appears as $E^1$-term of certain spectral sequence converging to $H_* (K)$. The main result is that these two Poisson structures are the same.

We compute the first non-trivial example of the Poisson bracket. We show that this gives a first example of the homology class of $K$ which does not directly correspond to any chord diagrams.

We study the homological properties of the space $K$ of (framed) long knots in $\\R^n$, $n>3$, in particular its Poisson algebra structures.

We had known two kinds of Poisson structures, both of which are based on the action of little disks operad. One definition is via the action on the space $K$. Another comes from the action of chains of little disks on the Hochschild complex of an operad, which appears as $E^1$-term of certain spectral sequence converging to $H_* (K)$. The main result is that these two Poisson structures are the same.

We compute the first non-trivial example of the Poisson bracket. We show that this gives a first example of the homology class of $K$ which does not directly correspond to any chord diagrams.

**吉田 享平**(東京大学大学院数理科学研究科) 17:30-18:30On projections of pseudo-ribbon sphere-links

[ Abstract ]

Suppose $F$ is an embedded closed surface in $R^4$.

We call $F$ a pseudo-ribbon surface link

if its projection is an immersion of $F$ into $R^3$

whose self-intersection set $\\Gamma(F)$ consists of disjointly embedded circles.

H. Aiso classified pseudo-ribbon sphere-knots ($F$ is a sphere.)

when $\\Gamma(F)$ consists of less than 6 circles.

We classify pseudo-ribbon sphere-links

when $F$ is two spheres and $\\Gamma(F)$ consists of less than 7 circles.

Suppose $F$ is an embedded closed surface in $R^4$.

We call $F$ a pseudo-ribbon surface link

if its projection is an immersion of $F$ into $R^3$

whose self-intersection set $\\Gamma(F)$ consists of disjointly embedded circles.

H. Aiso classified pseudo-ribbon sphere-knots ($F$ is a sphere.)

when $\\Gamma(F)$ consists of less than 6 circles.

We classify pseudo-ribbon sphere-links

when $F$ is two spheres and $\\Gamma(F)$ consists of less than 7 circles.

### 2006/12/12

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

Thom polynomials for maps of curves with isolated singularities

(joint with S. Lando)

**Maxim Kazarian**(Steklov Math. Institute)Thom polynomials for maps of curves with isolated singularities

(joint with S. Lando)

[ Abstract ]

Thom (residual) polynomials in characteristic classes are used in

the analysis of geometry of functional spaces. They serve as a

tool in description of classes Poincar\\'e dual to subvarieties of

functions of prescribed types. We give explicit universal

expressions for residual polynomials in spaces of functions on

complex curves having isolated singularities and

multisingularities, in terms of few characteristic classes. These

expressions lead to a partial explicit description of a

stratification of Hurwitz spaces.

Thom (residual) polynomials in characteristic classes are used in

the analysis of geometry of functional spaces. They serve as a

tool in description of classes Poincar\\'e dual to subvarieties of

functions of prescribed types. We give explicit universal

expressions for residual polynomials in spaces of functions on

complex curves having isolated singularities and

multisingularities, in terms of few characteristic classes. These

expressions lead to a partial explicit description of a

stratification of Hurwitz spaces.

### 2006/11/28

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

The Yamabe constants of infinite coverings and a positive mass theorem

**芥川 和雄**(東京理科大学理工学部)The Yamabe constants of infinite coverings and a positive mass theorem

[ Abstract ]

The {\\it Yamabe constant} $Y(M, C)$ of a given closed conformal manifold

$(M, C)$ is defined by the infimum of

the normalized total-scalar-curavarure functional $E$

among all metrics in $C$.

The study of the second variation of this functional $E$ led O.Kobayashi and Schoen

to independently introduce a natural differential-topological invariant $Y(M)$,

which is obtained by taking the supremum of $Y(M, C)$ over the space of all conformal classes.

This invariant $Y(M)$ is called the {\\it Yamabe invariant} of $M$.

For the study of the Yamabe invariant,

the relationship between $Y(M, C)$ and those of its conformal coverings

is important, the case when $Y(M, C)> 0$ particularly.

When $Y(M, C) \\leq 0$, by the uniqueness of unit-volume constant scalar curvature metrics in $C$,

the desired relation is clear.

When $Y(M, C) > 0$, such a uniqueness does not hold.

However, Aubin proved that $Y(M, C)$ is strictly less than

the Yamabe constant of any of its non-trivial {\\it finite} conformal coverings,

called {\\it Aubin's Lemma}.

In this talk, we generalize this lemma to the one for the Yamabe constant of

any $(M_{\\infty}, C_{\\infty})$ of its {\\it infinite} conformal coverings,

under a certain topological condition on the relation between $\\pi_1(M)$ and $\\pi_1(M_{\\infty})$.

For the proof of this, we aslo establish a version of positive mass theorem

for a specific class of asymptotically flat manifolds with singularities.

The {\\it Yamabe constant} $Y(M, C)$ of a given closed conformal manifold

$(M, C)$ is defined by the infimum of

the normalized total-scalar-curavarure functional $E$

among all metrics in $C$.

The study of the second variation of this functional $E$ led O.Kobayashi and Schoen

to independently introduce a natural differential-topological invariant $Y(M)$,

which is obtained by taking the supremum of $Y(M, C)$ over the space of all conformal classes.

This invariant $Y(M)$ is called the {\\it Yamabe invariant} of $M$.

For the study of the Yamabe invariant,

the relationship between $Y(M, C)$ and those of its conformal coverings

is important, the case when $Y(M, C)> 0$ particularly.

When $Y(M, C) \\leq 0$, by the uniqueness of unit-volume constant scalar curvature metrics in $C$,

the desired relation is clear.

When $Y(M, C) > 0$, such a uniqueness does not hold.

However, Aubin proved that $Y(M, C)$ is strictly less than

the Yamabe constant of any of its non-trivial {\\it finite} conformal coverings,

called {\\it Aubin's Lemma}.

In this talk, we generalize this lemma to the one for the Yamabe constant of

any $(M_{\\infty}, C_{\\infty})$ of its {\\it infinite} conformal coverings,

under a certain topological condition on the relation between $\\pi_1(M)$ and $\\pi_1(M_{\\infty})$.

For the proof of this, we aslo establish a version of positive mass theorem

for a specific class of asymptotically flat manifolds with singularities.

### 2006/11/14

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

High-codimensional knots spun about manifolds

**高瀬将道**(信州大学理学部)High-codimensional knots spun about manifolds

[ Abstract ]

The spinning describes several methods of constructing higher-dimensional knots from lower-dimensional knots.

The original spinning (Emil Artin, 1925) has been generalized in various ways. Using one of the most generalized forms of spinning, called "deform-spinning about a submanifold" (Dennis Roseman, 1989), we analyze in a geometric way Haefliger's smoothly knotted (4k-1)-spheres in the 6k-sphere.

The spinning describes several methods of constructing higher-dimensional knots from lower-dimensional knots.

The original spinning (Emil Artin, 1925) has been generalized in various ways. Using one of the most generalized forms of spinning, called "deform-spinning about a submanifold" (Dennis Roseman, 1989), we analyze in a geometric way Haefliger's smoothly knotted (4k-1)-spheres in the 6k-sphere.

### 2006/11/10

17:40-19:00 Room #118 (Graduate School of Math. Sci. Bldg.)

WRT invariant for Seifert manifolds and modular forms

**樋上和弘**(東京大学大学院理学系研究科 物理)WRT invariant for Seifert manifolds and modular forms

[ Abstract ]

We study the SU(2) Witten-Reshetikhin-Turaev invariant for Seifert manifold. We disuss a relationship with the Eichler integral of half-integral modular form and Ramanujan mock theta functions.

We study the SU(2) Witten-Reshetikhin-Turaev invariant for Seifert manifold. We disuss a relationship with the Eichler integral of half-integral modular form and Ramanujan mock theta functions.

### 2006/10/31

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

Unsmoothable group actions on elliptic surfaces

**中村 信裕**(東京大学大学院数理科学研究科)Unsmoothable group actions on elliptic surfaces

[ Abstract ]

Let G be a cyclic group of order 3,5 or 7.

We prove the existence of locally linear G-actions on elliptic surfaces which can not be realized by smooth actions with respect to specific smooth structures.

To prove this, we give constraints on smooth actions by using gauge theory.

In fact, we use a mod p vanishing theorem on Seiberg-Witten invariants, which was originally proved by F.Fang.

We give a geometric alternative proof of this, which enables us to extend the theorem.

Let G be a cyclic group of order 3,5 or 7.

We prove the existence of locally linear G-actions on elliptic surfaces which can not be realized by smooth actions with respect to specific smooth structures.

To prove this, we give constraints on smooth actions by using gauge theory.

In fact, we use a mod p vanishing theorem on Seiberg-Witten invariants, which was originally proved by F.Fang.

We give a geometric alternative proof of this, which enables us to extend the theorem.

### 2006/10/24

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

A review of crossed G-structures

**Marco Zunino**(JSPS, University of Tokyo)A review of crossed G-structures

[ Abstract ]

We present the definition of "crossed structures" as introduced by Turaev and others a few years ago. One of the original motivations in the introduction of these structures and of the related notion of a "Homotopy Quantum Field Theory" (HQFT) was the extension of Reshetikhin-Turaev invariants to the case of flat principal bundles on 3-manifolds. We resume both this aspect of the theory and other applications in both algebra and topology and we present our results on the algebraic structures involved.

We present the definition of "crossed structures" as introduced by Turaev and others a few years ago. One of the original motivations in the introduction of these structures and of the related notion of a "Homotopy Quantum Field Theory" (HQFT) was the extension of Reshetikhin-Turaev invariants to the case of flat principal bundles on 3-manifolds. We resume both this aspect of the theory and other applications in both algebra and topology and we present our results on the algebraic structures involved.

### 2006/10/17

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

Networking Seifert Fibered Surgeries on Knots (joint work with Katura Miyazaki and Kimihiko Motegi)

**Arnaud Deruelle**(University of Tokyo)Networking Seifert Fibered Surgeries on Knots (joint work with Katura Miyazaki and Kimihiko Motegi)

[ Abstract ]

We define a Seifert Surgery Network which consists of integral Dehn surgeries on knots yielding Seifert fiber spaces;here we allow Seifert fiber space with a fiber of index zero as degenerate cases. Then we establish some fundamental properties of the network. Using the notion of the network, we will clarify relationships among known Seifert surgeries. In particular, we demonstrate that many Seifert surgeries are closely connected to those on torus knots in Seifert Surgery Network. Our study suggests that the network enables us to make a global picture of Seifert surgeries.

We define a Seifert Surgery Network which consists of integral Dehn surgeries on knots yielding Seifert fiber spaces;here we allow Seifert fiber space with a fiber of index zero as degenerate cases. Then we establish some fundamental properties of the network. Using the notion of the network, we will clarify relationships among known Seifert surgeries. In particular, we demonstrate that many Seifert surgeries are closely connected to those on torus knots in Seifert Surgery Network. Our study suggests that the network enables us to make a global picture of Seifert surgeries.

### 2006/10/10

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

Estimating Lusternik-Schnirelmann Category for Foliations:A Survey of Available Techniques

**Elmar Vogt**(Frie Universitat Berlin)Estimating Lusternik-Schnirelmann Category for Foliations:A Survey of Available Techniques

[ Abstract ]

The Lusternik-Schnirelmann category of a space $X$ is the smallest number $r$ such that $X$ can be covered by $r + 1$ open sets which are contractible in $X$.For foliated manifolds there are several notions generalizing this concept, all of them due

to Helen Colman. We are mostly concerned with the concept of tangential Lusternik-Schnirelmann category (tangential LS-category). Here one requires a covering by open sets $U$ with the following property. There is a leafwise homotopy starting with the inclusion of $U$ and ending in a map that throws for each leaf $F$ of the foliation each component of $U \\cap F$ onto a single point. A leafwise homotopy is a homotopy moving points only inside leaves. Rather than presenting the still very few results obtained about the LS category of foliations, we survey techniques, mostly quite elementary, to estimate the tangential LS-category from below and above.

The Lusternik-Schnirelmann category of a space $X$ is the smallest number $r$ such that $X$ can be covered by $r + 1$ open sets which are contractible in $X$.For foliated manifolds there are several notions generalizing this concept, all of them due

to Helen Colman. We are mostly concerned with the concept of tangential Lusternik-Schnirelmann category (tangential LS-category). Here one requires a covering by open sets $U$ with the following property. There is a leafwise homotopy starting with the inclusion of $U$ and ending in a map that throws for each leaf $F$ of the foliation each component of $U \\cap F$ onto a single point. A leafwise homotopy is a homotopy moving points only inside leaves. Rather than presenting the still very few results obtained about the LS category of foliations, we survey techniques, mostly quite elementary, to estimate the tangential LS-category from below and above.

### 2006/07/24

16:30-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Invariant foliations in hyperbolic dynamics:

Smoothness and smooth equivalence

http://faculty.ms.u-tokyo.ac.jp/~topology/

**Boris Hasselblatt**(Tufts University)Invariant foliations in hyperbolic dynamics:

Smoothness and smooth equivalence

[ Abstract ]

The stable and unstable leaves of a hyperbolic dynamical system are smooth and form a continuous foliation. Smoothness of this foliation sometimes constrains the topological type of the foliation, other times restricts at least the smooth equivalence class of the dynamical system, or for geodesic flows, the type of the underlying manifold. I will present a broad introduction as well as recent work, work in progress, and open problems.

[ Reference URL ]The stable and unstable leaves of a hyperbolic dynamical system are smooth and form a continuous foliation. Smoothness of this foliation sometimes constrains the topological type of the foliation, other times restricts at least the smooth equivalence class of the dynamical system, or for geodesic flows, the type of the underlying manifold. I will present a broad introduction as well as recent work, work in progress, and open problems.

http://faculty.ms.u-tokyo.ac.jp/~topology/

### 2006/07/11

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

全葉層の存在について(浅岡正幸,Emmanuel Dufraineとの共同研究)

http://faculty.ms.u-tokyo.ac.jp/~topology/

**野田健夫**(秋田大学工学資源学部)全葉層の存在について(浅岡正幸,Emmanuel Dufraineとの共同研究)

[ Abstract ]

n次元多様体上のn個の余次元1葉層構造の組で、n個の葉層構造の接空間の共通部分が各点で0になるものを全葉層と呼ぶ。3次元の場合においては任意の有向閉多様体上に全葉層が存在することが Hardorpによって示されていた。3次元多様体上の全葉層をなす各々の葉層構造の接平面場は互いにホモトピックでありオイラー類が0になることが容易に分かるが、逆にオイラー類が0の平面場を与えたときそれを実現する全葉層が存在するかという問題が自然に生じる。

本講演ではこの問題に肯定的な解決をあたえる。

また、この結果の応用として双接触構造、すなわち横断的に交わる正と負の接触構造の組の存在問題にも触れたい。

[ Reference URL ]n次元多様体上のn個の余次元1葉層構造の組で、n個の葉層構造の接空間の共通部分が各点で0になるものを全葉層と呼ぶ。3次元の場合においては任意の有向閉多様体上に全葉層が存在することが Hardorpによって示されていた。3次元多様体上の全葉層をなす各々の葉層構造の接平面場は互いにホモトピックでありオイラー類が0になることが容易に分かるが、逆にオイラー類が0の平面場を与えたときそれを実現する全葉層が存在するかという問題が自然に生じる。

本講演ではこの問題に肯定的な解決をあたえる。

また、この結果の応用として双接触構造、すなわち横断的に交わる正と負の接触構造の組の存在問題にも触れたい。

http://faculty.ms.u-tokyo.ac.jp/~topology/

### 2006/07/04

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

Amalgams: a machinery of the modern theory of finite groups

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/~topology/

**Alexander A. Ivanov**(Imperial College (London))Amalgams: a machinery of the modern theory of finite groups

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/~topology/

### 2006/06/27

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

Lorentzian foliations on 3-manifolds

**Cedric Tarquini**(Ecole Nomale Superieure of Lyon)Lorentzian foliations on 3-manifolds

[ Abstract ]

a joint work with C. Boubel (Ecole Nomale Superieure of Lyon) and P. Mounoud (University of Bordeaux 1 sciences and technologies)

The aim of this work is to give a classification of transversely Lorentzian one dimensional foliations on compact manifolds of dimension three. There are the foliations which admit a transverse pseudo-Riemanniann metric of index one. It is the Lorentzian analogue of the better known Riemannian foliations and they still have rigid transverse geometry.

The Riemannian case was listed by Y. Carriere and we will see that the Lorentzian one is very different and much more complicated to classify. The difference comes form the fact that the completness of the transverse structure, which is automatic in the Riemannian case, is a very strong hypothesis for a transverse Lorentzian foliation.

We will give a classification of complete Lorentzian foliations and some examples which are not complete. As a natural corollary of this classification we will list the codimension one timelike geodesically complete totally geodesic foliations of Lorentzian compact three manifolds.

a joint work with C. Boubel (Ecole Nomale Superieure of Lyon) and P. Mounoud (University of Bordeaux 1 sciences and technologies)

The aim of this work is to give a classification of transversely Lorentzian one dimensional foliations on compact manifolds of dimension three. There are the foliations which admit a transverse pseudo-Riemanniann metric of index one. It is the Lorentzian analogue of the better known Riemannian foliations and they still have rigid transverse geometry.

The Riemannian case was listed by Y. Carriere and we will see that the Lorentzian one is very different and much more complicated to classify. The difference comes form the fact that the completness of the transverse structure, which is automatic in the Riemannian case, is a very strong hypothesis for a transverse Lorentzian foliation.

We will give a classification of complete Lorentzian foliations and some examples which are not complete. As a natural corollary of this classification we will list the codimension one timelike geodesically complete totally geodesic foliations of Lorentzian compact three manifolds.

### 2006/06/13

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

A note on C1-moves

**田中 心**(東京大学大学院数理科学研究科)A note on C1-moves

[ Abstract ]

鎌田氏によりチャートという概念が定義された。これは二次元円板上の 有向ラベル付きグラフであり、二次元ブレイドを記述する際に用いられる。 彼はチャートに対してC変形と呼ばれる三種類の変形(C1変形、C2変形、C3変形) を定義し、曲面ブレイドの同値類とチャートのC変形同値類の間に一対一対応が ある事を示した。 カーター氏と斎藤氏は、任意のC1変形は七種類の基本C1変形の列で得られる 事を示したが、その証明には曖昧な部分がある事が知られていた。本講演では 彼らとは異なるアプローチにより、彼らの主張に対して正しい証明を与える。

鎌田氏によりチャートという概念が定義された。これは二次元円板上の 有向ラベル付きグラフであり、二次元ブレイドを記述する際に用いられる。 彼はチャートに対してC変形と呼ばれる三種類の変形(C1変形、C2変形、C3変形) を定義し、曲面ブレイドの同値類とチャートのC変形同値類の間に一対一対応が ある事を示した。 カーター氏と斎藤氏は、任意のC1変形は七種類の基本C1変形の列で得られる 事を示したが、その証明には曖昧な部分がある事が知られていた。本講演では 彼らとは異なるアプローチにより、彼らの主張に対して正しい証明を与える。

### 2006/06/06

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

Thurston's inequality for a foliation with Reeb components

**三好 重明**(中央大学理工学部)Thurston's inequality for a foliation with Reeb components

[ Abstract ]

The Euler class of a Reebless foliation or a tight contact structure on a closed 3-manifold satisfies Thurston's inequality, i.e. its (dual) Thurston norm is less than or equal to 1. It should be significant to study Thurston's inequality in both of foliation theory and contact topology. We investigate conditions for a spinnable foliation one of which assures that Thurston's inequality holds and also another of which implies the violation of the inequality.

The Euler class of a Reebless foliation or a tight contact structure on a closed 3-manifold satisfies Thurston's inequality, i.e. its (dual) Thurston norm is less than or equal to 1. It should be significant to study Thurston's inequality in both of foliation theory and contact topology. We investigate conditions for a spinnable foliation one of which assures that Thurston's inequality holds and also another of which implies the violation of the inequality.

### 2006/05/30

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

自由群の自己同型群のJohnson準同型の余核について

**佐藤 隆夫**(東京大学大学院数理科学研究科)自由群の自己同型群のJohnson準同型の余核について

[ Abstract ]

本講演では,まず次数が2,3の場合に自由群の自己同型群の Johnson準同型の余核の構造を決定する.さらに,次数1の元たちが生成する部分に定義域を制限することで,奇数次のJohnson準同型の全射性に関して新しい障害が得られたことを紹介する.

本講演では,まず次数が2,3の場合に自由群の自己同型群の Johnson準同型の余核の構造を決定する.さらに,次数1の元たちが生成する部分に定義域を制限することで,奇数次のJohnson準同型の全射性に関して新しい障害が得られたことを紹介する.

### 2006/05/23

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

On crossed homomorphisms on symplectic mapping class groups

**笠川 良司**(日本大学理工学部)On crossed homomorphisms on symplectic mapping class groups

[ Abstract ]

We consider a symplectic manifold M. For a relation between Chern classes of M and the cohomology class of the symplectic form, we construct a crossed homomorphism on the symplectomorphism group of M with values in the cohomology group of M. We show an application of it to the flux homomorphism. Then we see that it induces a one on the symplectic mapping class group of M and show a nontrivial example of it.

We consider a symplectic manifold M. For a relation between Chern classes of M and the cohomology class of the symplectic form, we construct a crossed homomorphism on the symplectomorphism group of M with values in the cohomology group of M. We show an application of it to the flux homomorphism. Then we see that it induces a one on the symplectic mapping class group of M and show a nontrivial example of it.

### 2006/05/16

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Fundamental groups of complements to complex hypersurfaces

**Laurentiu Maxim**(University of Illinois at Chicago)Fundamental groups of complements to complex hypersurfaces

[ Abstract ]

I will survey various Alexander-type invariants of hypersurface complements, with an emphasis on obstructions on the type of groups that can arise as fundamental groups of complements to affine hypersurfaces.

I will survey various Alexander-type invariants of hypersurface complements, with an emphasis on obstructions on the type of groups that can arise as fundamental groups of complements to affine hypersurfaces.

### 2006/04/25

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

Counting closed orbits and flow lines via Heegaard splittings

**合田 洋**(東京農工大学)Counting closed orbits and flow lines via Heegaard splittings

[ Abstract ]

Let K be a fibred knot in the 3-sphere. It is known that the Alexander polynomial of K is essentially equal to a Lefschetz zeta function obtained from the monodromy map of the fibre structure. In this talk, we discuss the non-fibred knot case. We introduce "monodromy matrix" by making use of Heegaard splitting for sutured manifolds of a knot K, and then observe a method of counting closed orbits and flow lines, which gives the Alexander polynomial of K. This observation is based on works of Donaldson and Mark. (joint work with Hiroshi Matsuda and Andrei Pajitnov)

Let K be a fibred knot in the 3-sphere. It is known that the Alexander polynomial of K is essentially equal to a Lefschetz zeta function obtained from the monodromy map of the fibre structure. In this talk, we discuss the non-fibred knot case. We introduce "monodromy matrix" by making use of Heegaard splitting for sutured manifolds of a knot K, and then observe a method of counting closed orbits and flow lines, which gives the Alexander polynomial of K. This observation is based on works of Donaldson and Mark. (joint work with Hiroshi Matsuda and Andrei Pajitnov)

### 2006/04/18

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

Topology of words

**Vladimir Turaev**(Univ. Louis Pasteur Strasbourg)Topology of words

[ Abstract ]

There is a parallel between words, defined as finite sequences of letters, and curves on surfaces. This allows to treat words as geometric objects and to analyze them using techniques from low-dimensional topology. I will discuss basic ideas in this direction and the resulting topological invariants of words.

There is a parallel between words, defined as finite sequences of letters, and curves on surfaces. This allows to treat words as geometric objects and to analyze them using techniques from low-dimensional topology. I will discuss basic ideas in this direction and the resulting topological invariants of words.

### 2006/04/11

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

Homotopy actions, cyclic maps and their Eckmann-Hilton duals.

**Martin Arkowitz**(Dartmouth College)Homotopy actions, cyclic maps and their Eckmann-Hilton duals.

[ Abstract ]

We study the homotopy action of a based space A on a based space X. The resulting map A--->X is called cyclic. We classify actions on an H-space which are compatible with the H-structure. In the dual case we study coactions X--->X v B and the resulting cocyclic map X--->B. We relate the cocyclicity of a map to the Lusternik-Schnirelmann category of the map.

We study the homotopy action of a based space A on a based space X. The resulting map A--->X is called cyclic. We classify actions on an H-space which are compatible with the H-structure. In the dual case we study coactions X--->X v B and the resulting cocyclic map X--->B. We relate the cocyclicity of a map to the Lusternik-Schnirelmann category of the map.