## Tuesday Seminar on Topology

Seminar information archive ～06/22｜Next seminar｜Future seminars 06/23～

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**

### 2007/10/09

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

Classification of codimension-one locally free actions of the affine group of the real line.

**浅岡 正幸**(京都大学大学院理学研究科)Classification of codimension-one locally free actions of the affine group of the real line.

[ Abstract ]

By GA, we denote the group of affine and orientation-preserving transformations

of the real line. In this talk, I will report on classification of locally free action of

GA on closed three manifolds, which I obtained recently. In 1979, E.Ghys proved

that if such an action preserves a volume, then it is smoothly conjugate to a homogeneous action. However, it was unknown whether non-homogeneous action exists. As a consequence of the classification, we will see that the unit tangent bundle of a closed surface of higher genus admits a finite-parameter family of

non-homogeneous actions.

By GA, we denote the group of affine and orientation-preserving transformations

of the real line. In this talk, I will report on classification of locally free action of

GA on closed three manifolds, which I obtained recently. In 1979, E.Ghys proved

that if such an action preserves a volume, then it is smoothly conjugate to a homogeneous action. However, it was unknown whether non-homogeneous action exists. As a consequence of the classification, we will see that the unit tangent bundle of a closed surface of higher genus admits a finite-parameter family of

non-homogeneous actions.

### 2007/07/17

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

Orbifold Cohomology of Wreath Product Orbifolds and

Cohomological HyperKahler Resolution Conjecture

**松村 朝雄**(東京大学大学院数理科学研究科)Orbifold Cohomology of Wreath Product Orbifolds and

Cohomological HyperKahler Resolution Conjecture

[ Abstract ]

Chen-Ruan orbifold cohomology ring was introduced in 2000 as

the degree zero genus zero orbifold Gromov-Witten invariants with

three marked points. We will review its construction in the case of

global quotient orbifolds, following Fantechi-Gottsche and

Jarvis-Kaufmann-Kimura. We will describe the orbifold cohomology of

wreath product orbifolds and explain its application to Ruan's

cohomological hyperKahler resolution conjecture.

Chen-Ruan orbifold cohomology ring was introduced in 2000 as

the degree zero genus zero orbifold Gromov-Witten invariants with

three marked points. We will review its construction in the case of

global quotient orbifolds, following Fantechi-Gottsche and

Jarvis-Kaufmann-Kimura. We will describe the orbifold cohomology of

wreath product orbifolds and explain its application to Ruan's

cohomological hyperKahler resolution conjecture.

### 2007/07/10

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

Combable functions, quasimorphisms, and the central limit theorem

(joint with Koji Fujiwara)

**Danny C. Calegari**(California Institute of Technology)Combable functions, quasimorphisms, and the central limit theorem

(joint with Koji Fujiwara)

[ Abstract ]

Quasimorphisms on groups are dual to stable commutator length,

and detect extremal phenomena in topology and dynamics. In typical groups

(even in a free group) stable commutator length is very difficult to

calculate, because the space of quasimorphisms is too large to study

directly without adding more structure.

In this talk, we show that a large class of quasimorphisms - the so-called

"counting quasimorphisms" on word-hyperbolic groups - can be effectively

described using simple machines called finite state automata. From this,

and from the ergodic theory of finite directed graphs, one can deduce a

number of properties about the statistical distribution of the values of a

counting quasimorphism on elements of the group.

Quasimorphisms on groups are dual to stable commutator length,

and detect extremal phenomena in topology and dynamics. In typical groups

(even in a free group) stable commutator length is very difficult to

calculate, because the space of quasimorphisms is too large to study

directly without adding more structure.

In this talk, we show that a large class of quasimorphisms - the so-called

"counting quasimorphisms" on word-hyperbolic groups - can be effectively

described using simple machines called finite state automata. From this,

and from the ergodic theory of finite directed graphs, one can deduce a

number of properties about the statistical distribution of the values of a

counting quasimorphism on elements of the group.

### 2007/07/03

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

Two invariants of pseudo--Anosov mapping classes: hyperbolic volume vs dilatation

(joint work with Mitsuhiko Takasawa)

**金 英子**(東京工業大学情報理工学研究科)Two invariants of pseudo--Anosov mapping classes: hyperbolic volume vs dilatation

(joint work with Mitsuhiko Takasawa)

[ Abstract ]

We concern two invariants of pseudo-Anosov mapping classes.

One is the dilatation of pseudo--Anosov maps and the other is the volume

of mapping tori. To study how two invariants are related, fixing a surface

we represent a mapping class by using the standard generator set and compute

these two for all pseudo--Anosov mapping classes with up to some word length.

In the talk, we observe two properties:

(1) The ratio of the topological entropy (i.e. logarithm of the dilatation) to

the volume is bounded from below by some positive constant which only

depends on the surface.

(2) The conjugacy class having the minimal dilatation reaches the minimal volume.

On the observation (1), in case of the mapping class group of a once--punctured

torus, we give a concrete lower bound of the ratio.

We concern two invariants of pseudo-Anosov mapping classes.

One is the dilatation of pseudo--Anosov maps and the other is the volume

of mapping tori. To study how two invariants are related, fixing a surface

we represent a mapping class by using the standard generator set and compute

these two for all pseudo--Anosov mapping classes with up to some word length.

In the talk, we observe two properties:

(1) The ratio of the topological entropy (i.e. logarithm of the dilatation) to

the volume is bounded from below by some positive constant which only

depends on the surface.

(2) The conjugacy class having the minimal dilatation reaches the minimal volume.

On the observation (1), in case of the mapping class group of a once--punctured

torus, we give a concrete lower bound of the ratio.

### 2007/06/12

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

The Kodaira dimension of symplectic 4-manifolds

**Tian-Jun Li**(University of Minnesota)The Kodaira dimension of symplectic 4-manifolds

[ Abstract ]

Various results and questions about symplectic4-manifolds can be

formulated in terms of the notion of the Kodaira dimension. In particular,

we will discuss the classification and the geography problems. It is interesting

to understand how it behaves undersome basic constructions.Time permitting

we will discuss the symplectic birational aspect of this notion and speculate

how to extend it to higher dimensional manifolds.

Various results and questions about symplectic4-manifolds can be

formulated in terms of the notion of the Kodaira dimension. In particular,

we will discuss the classification and the geography problems. It is interesting

to understand how it behaves undersome basic constructions.Time permitting

we will discuss the symplectic birational aspect of this notion and speculate

how to extend it to higher dimensional manifolds.

### 2007/06/05

17:00-18:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Symplectic mapping classes and fillings

**Emmanuel Giroux**(ENS Lyon)Symplectic mapping classes and fillings

[ Abstract ]

We will describe a joint work in progress with Paul Biran in

which contact geometry is combined with properties of Lagrangian manifolds

in subcritical Stein domains to obtain nontrivaility results for symplectic

mapping classes.

We will describe a joint work in progress with Paul Biran in

which contact geometry is combined with properties of Lagrangian manifolds

in subcritical Stein domains to obtain nontrivaility results for symplectic

mapping classes.

### 2007/05/15

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

Kontsevich's characteristic classes for higher dimensional homology sphere bundles

**渡邉 忠之**(京都大学数理解析研究所)Kontsevich's characteristic classes for higher dimensional homology sphere bundles

[ Abstract ]

As an analogue of the perturbative Chern-Simons theory, Maxim Kontsevich

constructed universal characteristic classes of smooth fiber bundles with fiber

diffeomorphic to a singularly framed odd dimensional homology sphere.

In this talk, I will give a sketch proof of our result on non-triviality of the

Kontsevich classes for 7-dimensional homology sphere bundles.

As an analogue of the perturbative Chern-Simons theory, Maxim Kontsevich

constructed universal characteristic classes of smooth fiber bundles with fiber

diffeomorphic to a singularly framed odd dimensional homology sphere.

In this talk, I will give a sketch proof of our result on non-triviality of the

Kontsevich classes for 7-dimensional homology sphere bundles.

### 2007/05/08

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

On the vanishing of the Rohlin invariant

**森山 哲裕**(東京大学大学院数理科学研究科)On the vanishing of the Rohlin invariant

[ Abstract ]

The vanishing of the Rohlin invariant of an amphichiral integral

homology $3$-sphere $M$ (i.e. $M \\cong -M$) is a natural consequence

of some elementary properties of the Casson invariant. In this talk, we

give a new direct (and more elementary) proof of this vanishing

property. The main idea comes from the definition of the degree 1

part of the Kontsevich-Kuperberg-Thurston invariant, and we progress

by constructing some $7$-dimensional manifolds in which $M$ is embedded.

The vanishing of the Rohlin invariant of an amphichiral integral

homology $3$-sphere $M$ (i.e. $M \\cong -M$) is a natural consequence

of some elementary properties of the Casson invariant. In this talk, we

give a new direct (and more elementary) proof of this vanishing

property. The main idea comes from the definition of the degree 1

part of the Kontsevich-Kuperberg-Thurston invariant, and we progress

by constructing some $7$-dimensional manifolds in which $M$ is embedded.

### 2007/04/24

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

Realization of twisted $K$-theory and

finite-dimensional approximation of Fredholm operators

**五味 清紀**(東京大学大学院数理科学研究科)Realization of twisted $K$-theory and

finite-dimensional approximation of Fredholm operators

[ Abstract ]

A problem in twisted $K$-theory is to realize twisted $K$-groups generally by means of finite-dimensional geometric objects, like vector bundles. I would like to talk about an approach toward the problem by means of Mikio Furuta's generalized vector bundles. By using a twisted version of the generalized vector bundle and a finite-dimensional approximation of Fredholm operators, I construct a group into which there exists a natural injection from the twisted $K$-group twisted by any third integral cohomology class.

A problem in twisted $K$-theory is to realize twisted $K$-groups generally by means of finite-dimensional geometric objects, like vector bundles. I would like to talk about an approach toward the problem by means of Mikio Furuta's generalized vector bundles. By using a twisted version of the generalized vector bundle and a finite-dimensional approximation of Fredholm operators, I construct a group into which there exists a natural injection from the twisted $K$-group twisted by any third integral cohomology class.

### 2007/04/17

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

Existence Problem of Compact Locally Symmetric Spaces

**小林 俊行**(東京大学大学院数理科学研究科)Existence Problem of Compact Locally Symmetric Spaces

[ Abstract ]

The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particularly in Riemannian geometry. In contrast, in areas such as Lorentz geometry, familiar to us as the space-time of relativity theory, and more generally in pseudo-Riemannian geometry, as well as in various other kinds of geometry (symplectic, complex geometry, ...), surprising little is known about global properties of the geometry even if we impose a locally homogeneous structure.

In this talk, I will give a survey on the recent developments on the question about how the local geometric structure affects the global nature of non-Riemannian manifolds with emphasis on the existence problem of compact models of locally symmetric spaces.

The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particularly in Riemannian geometry. In contrast, in areas such as Lorentz geometry, familiar to us as the space-time of relativity theory, and more generally in pseudo-Riemannian geometry, as well as in various other kinds of geometry (symplectic, complex geometry, ...), surprising little is known about global properties of the geometry even if we impose a locally homogeneous structure.

In this talk, I will give a survey on the recent developments on the question about how the local geometric structure affects the global nature of non-Riemannian manifolds with emphasis on the existence problem of compact models of locally symmetric spaces.

### 2007/01/30

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

Elliptic genera and some finite groups

**John F. Duncan**(Harvard University)Elliptic genera and some finite groups

[ Abstract ]

Recent developments in the representation theory of sporadic groups

suggest new formulations of `moonshine' in which Jacobi forms take on the

role played by modular forms in the monstrous case. On the other hand,

Jacobi forms arise naturally in the study of elliptic genera. We review

the use of vertex algebra as a tool for constructing the elliptic genus of

a suitable vector bundle, and illustrate connections between this and

vertex algebraic representations of certain sporadic simple groups.

Recent developments in the representation theory of sporadic groups

suggest new formulations of `moonshine' in which Jacobi forms take on the

role played by modular forms in the monstrous case. On the other hand,

Jacobi forms arise naturally in the study of elliptic genera. We review

the use of vertex algebra as a tool for constructing the elliptic genus of

a suitable vector bundle, and illustrate connections between this and

vertex algebraic representations of certain sporadic simple groups.

### 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/