## Tuesday Seminar on Topology

Seminar information archive ～09/23｜Next seminar｜Future seminars 09/24～

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**

### 2008/10/28

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

Nonsmoothable group actions on spin 4-manifolds

**清野 和彦**(東京大学大学院数理科学研究科)Nonsmoothable group actions on spin 4-manifolds

[ Abstract ]

We call a locally linear group action on a topological manifold nonsmoothable

if the action is not smooth with respect to any possible smooth structure.

We show in this lecture that every closed, simply connected, spin topological 4-manifold

not homeomorphic to neither S^2\\times S^2 nor S^4 allows a nonsmoothable

group action of any cyclic group with sufficiently large prime order

which depends on the manifold.

We call a locally linear group action on a topological manifold nonsmoothable

if the action is not smooth with respect to any possible smooth structure.

We show in this lecture that every closed, simply connected, spin topological 4-manifold

not homeomorphic to neither S^2\\times S^2 nor S^4 allows a nonsmoothable

group action of any cyclic group with sufficiently large prime order

which depends on the manifold.

### 2008/10/21

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

On embeddings of 3-manifolds in 6-manifolds

**森山 哲裕**(東京大学大学院数理科学研究科)On embeddings of 3-manifolds in 6-manifolds

[ Abstract ]

In this talk, we give a simple axiomatic definition of an invariant of

smooth embeddings of 3-manifolds in 6-manifolds.

The axiom is expressed in terms of some cobordisms of pairs of manifolds of

dimensions 6 and 3 (equipped with some cohomology class of the complement) and

the signature of 4-manifolds.

We then show that our invariant gives a unified framework for some classical

invariants in low-dimensions (Haefliger invariant, Milnor's triple

linking number, Rokhlin invariant, Casson invariant,

Takase's invariant, Skopenkov's invariants).

In this talk, we give a simple axiomatic definition of an invariant of

smooth embeddings of 3-manifolds in 6-manifolds.

The axiom is expressed in terms of some cobordisms of pairs of manifolds of

dimensions 6 and 3 (equipped with some cohomology class of the complement) and

the signature of 4-manifolds.

We then show that our invariant gives a unified framework for some classical

invariants in low-dimensions (Haefliger invariant, Milnor's triple

linking number, Rokhlin invariant, Casson invariant,

Takase's invariant, Skopenkov's invariants).

### 2008/10/14

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

Pontrjagin-Thom maps and the Deligne-Mumford compactification

**Jeffrey Herschel Giansiracusa**(Oxford University)Pontrjagin-Thom maps and the Deligne-Mumford compactification

[ Abstract ]

An embedding f: M -> N produces, via a construction of Pontrjagin-Thom, a map from N to the Thom space of the normal bundle over M. If f is an arbitrary map then one instead gets a map from N to the infinite loop space of the Thom spectrum of the normal bundle of f. We extend this Pontrjagin-Thom construction of wrong-way maps to differentiable stacks and use it to produce interesting maps from the Deligne-Mumford compactification of the moduli space of curves to certain infinite loop spaces. We show that these maps are surjective on mod p homology in a range of degrees. We thus produce large new families of torsion cohomology classes on the Deligne-Mumford compactification.

An embedding f: M -> N produces, via a construction of Pontrjagin-Thom, a map from N to the Thom space of the normal bundle over M. If f is an arbitrary map then one instead gets a map from N to the infinite loop space of the Thom spectrum of the normal bundle of f. We extend this Pontrjagin-Thom construction of wrong-way maps to differentiable stacks and use it to produce interesting maps from the Deligne-Mumford compactification of the moduli space of curves to certain infinite loop spaces. We show that these maps are surjective on mod p homology in a range of degrees. We thus produce large new families of torsion cohomology classes on the Deligne-Mumford compactification.

### 2008/07/15

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

One-step Markov Theorem on exchange classes

**松田 浩**(広島大学大学院理学研究科)One-step Markov Theorem on exchange classes

[ Abstract ]

The main theorem in this talk claims the following.

``Let L_A and L_B denote a closed a-braid and a closed b-braid,

respectively, that represent one link type.

After at most (a^2 b^2)/4 exchange moves on L_A,

we can 'see' the pair of closed braids."

In this talk, we explain the main theorem in details, and

we present some applications.

In particular, we propose a strategy to construct an algorithm

that determines whether two links are ambient isotopic.

The main theorem in this talk claims the following.

``Let L_A and L_B denote a closed a-braid and a closed b-braid,

respectively, that represent one link type.

After at most (a^2 b^2)/4 exchange moves on L_A,

we can 'see' the pair of closed braids."

In this talk, we explain the main theorem in details, and

we present some applications.

In particular, we propose a strategy to construct an algorithm

that determines whether two links are ambient isotopic.

### 2008/07/08

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

Contact homology of left-handed stabilizations and connected sums

**Otto van Koert**(北海道大学大学院理学研究科, JSPS)Contact homology of left-handed stabilizations and connected sums

[ Abstract ]

In this talk, we will give a brief introduction to contact homology, an invariant of contact manifolds defined by counting holomorphic curves in the symplectization of a contact manifold. We shall show that certain left-handed stabilizations of contact open books have vanishing contact homology.

This is done by finding restrictions on the behavior of holomorphic curves in connected sums. An additional application of this behavior of holomorphic curves is a long exact sequence for linearized contact homology.

In this talk, we will give a brief introduction to contact homology, an invariant of contact manifolds defined by counting holomorphic curves in the symplectization of a contact manifold. We shall show that certain left-handed stabilizations of contact open books have vanishing contact homology.

This is done by finding restrictions on the behavior of holomorphic curves in connected sums. An additional application of this behavior of holomorphic curves is a long exact sequence for linearized contact homology.

### 2008/07/01

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

On the Johnson homomorphisms of the automorphism group of

a free metabelian group

**佐藤 隆夫**(大阪大学大学院理学研究科, JSPS)On the Johnson homomorphisms of the automorphism group of

a free metabelian group

[ Abstract ]

The main object of our research is the automorphism group of a

free group. To be brief, the Johnson homomorphisms are studied in order to

describe one by one approximations of the automorphism group of a free group

. They play important roles on the study of the homology groups of the autom

orphism group of a free group. In general, to determine their images are ver

y difficult problem. In this talk, we define the Johnson homomorphisms of th

e automorphism group of a free metabelian group, and determine their images.

Using these results, we can give a lower bound on the image of the Johnson

homomorphisms of the automorphism group of a free group.

The main object of our research is the automorphism group of a

free group. To be brief, the Johnson homomorphisms are studied in order to

describe one by one approximations of the automorphism group of a free group

. They play important roles on the study of the homology groups of the autom

orphism group of a free group. In general, to determine their images are ver

y difficult problem. In this talk, we define the Johnson homomorphisms of th

e automorphism group of a free metabelian group, and determine their images.

Using these results, we can give a lower bound on the image of the Johnson

homomorphisms of the automorphism group of a free group.

### 2008/06/24

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

On computing distances in the pants complex

http://www.is.titech.ac.jp/~kjshack5/FYEO.pdf

**Kenneth Shackleton**(東京工業大学, JSPS)On computing distances in the pants complex

[ Abstract ]

The pants complex is an accurate combinatorial

model for the Weil-Petersson metric (WP) on Teichmueller space

(Brock). One hopes that many of the geometric properties

of WP are accurately replicated in the pants complex, and

this is the source of many open questions. We compare these

in general, and then focus on the 5-holed sphere and the

2-holed torus, the first non-trivial surfaces. We arrive at

an algorithm for computing distances in the (1-skeleton of the)

pants complex of either surface.

[ Reference URL ]The pants complex is an accurate combinatorial

model for the Weil-Petersson metric (WP) on Teichmueller space

(Brock). One hopes that many of the geometric properties

of WP are accurately replicated in the pants complex, and

this is the source of many open questions. We compare these

in general, and then focus on the 5-holed sphere and the

2-holed torus, the first non-trivial surfaces. We arrive at

an algorithm for computing distances in the (1-skeleton of the)

pants complex of either surface.

http://www.is.titech.ac.jp/~kjshack5/FYEO.pdf

### 2008/06/17

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

Multiplier ideal sheaves and Futaki invariant on toric Fano manifolds.

**佐野 友二**(東京大学IPMU)Multiplier ideal sheaves and Futaki invariant on toric Fano manifolds.

[ Abstract ]

I would like to discuss the subvarieties cut off by the multiplier

ideal sheaves (MIS) and Futaki invariant on toric Fano manifolds.

Futaki invariant is one of the necessary conditions for the existence

of Kahler-Einstein metrics on Fano manifolds,

on the other hand MIS is one of the sufficient conditions introduced by Nadel.

Especially I would like to focus on the MIS related to the Monge-Ampere equation

for Kahler-Einstein metrics on non-KE toric Fano manifolds.

The motivation of this work comes from the investigation of the

relationship with slope stability

of polarized manifolds introduced by Ross and Thomas.

This talk will be based on a part of the joint work with Akito Futaki

(arXiv:0711.0614).

I would like to discuss the subvarieties cut off by the multiplier

ideal sheaves (MIS) and Futaki invariant on toric Fano manifolds.

Futaki invariant is one of the necessary conditions for the existence

of Kahler-Einstein metrics on Fano manifolds,

on the other hand MIS is one of the sufficient conditions introduced by Nadel.

Especially I would like to focus on the MIS related to the Monge-Ampere equation

for Kahler-Einstein metrics on non-KE toric Fano manifolds.

The motivation of this work comes from the investigation of the

relationship with slope stability

of polarized manifolds introduced by Ross and Thomas.

This talk will be based on a part of the joint work with Akito Futaki

(arXiv:0711.0614).

### 2008/06/03

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

On the geometry of certain slices of the character variety of a knot group

**山口 祥司**(東京大学大学院数理科学研究科)On the geometry of certain slices of the character variety of a knot group

[ Abstract ]

joint work with Fumikazu Nagasato (Meijo University)

This talk is concerned with certain subsets in the character variety of a knot group.

These subsets are called '"slices", which are defined as a level set of a regular function associated to a meridian of a knot.

They are related to character varieties for branched covers along the knot.

Some investigations indicate that an equivariant theory for a knot is connected to a theory for branched covers via slices, for example, the equivariant signature of a knot and the equivariant Casson invariant.

In this talk, we will construct a map from slices into the character varieties for branched covers and investigate the properties.

In particular, we focus on slices called "trace-free", which are used to define the Casson-Lin invariant, and the relation to the character variety for two--fold branched cover.

joint work with Fumikazu Nagasato (Meijo University)

This talk is concerned with certain subsets in the character variety of a knot group.

These subsets are called '"slices", which are defined as a level set of a regular function associated to a meridian of a knot.

They are related to character varieties for branched covers along the knot.

Some investigations indicate that an equivariant theory for a knot is connected to a theory for branched covers via slices, for example, the equivariant signature of a knot and the equivariant Casson invariant.

In this talk, we will construct a map from slices into the character varieties for branched covers and investigate the properties.

In particular, we focus on slices called "trace-free", which are used to define the Casson-Lin invariant, and the relation to the character variety for two--fold branched cover.

### 2008/05/20

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

Turaev-Viro TQFT splitting.

**Jer\^ome Petit**(東京工業大学, JSPS)Turaev-Viro TQFT splitting.

[ Abstract ]

The Turaev-Viro invariant is a 3-manifolds invariant. It is obtained in this way :

1) we use a combinatorial description of 3-manifolds, in this case it is : triangulation / Pachner moves

2) we define a scalar thanks to a categorical data (spherical category) and a topological data (triangulation)

3)we verify that the scalar is invariant under Pachner moves, then we obtain a 3-manifolds invariant.

The Turaev-Viro invariant can also be extended to a TQFT. Roughly speaking a TQFT is a data which assigns a finite dimensional vector space to every closed surface and a linear application to every 3-manifold with boundary.

In this talk, we will give a decomposition of the Turaev-Viro TQFT. More precisely, we decompose it into blocks. These blocks are given by a group associated to the spherical category which was used to construct the Turaev-Viro invariant. We will show that these blocks define a HQFT (roughly speaking a TQFT with an "homotopical data"). This HQFT is obtained from an homotopical invariant, which is the homotopical version of the Turaev-Viro invariant. Moreover this invariant can be used to obtain the homological Turaev-Viro invariant defined by Yetter.

The Turaev-Viro invariant is a 3-manifolds invariant. It is obtained in this way :

1) we use a combinatorial description of 3-manifolds, in this case it is : triangulation / Pachner moves

2) we define a scalar thanks to a categorical data (spherical category) and a topological data (triangulation)

3)we verify that the scalar is invariant under Pachner moves, then we obtain a 3-manifolds invariant.

The Turaev-Viro invariant can also be extended to a TQFT. Roughly speaking a TQFT is a data which assigns a finite dimensional vector space to every closed surface and a linear application to every 3-manifold with boundary.

In this talk, we will give a decomposition of the Turaev-Viro TQFT. More precisely, we decompose it into blocks. These blocks are given by a group associated to the spherical category which was used to construct the Turaev-Viro invariant. We will show that these blocks define a HQFT (roughly speaking a TQFT with an "homotopical data"). This HQFT is obtained from an homotopical invariant, which is the homotopical version of the Turaev-Viro invariant. Moreover this invariant can be used to obtain the homological Turaev-Viro invariant defined by Yetter.

### 2008/05/13

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

The problem of maximum Thurston--Bennequin number for knots

**Tamas Kalman**(東京大学大学院数理科学研究科, JSPS)The problem of maximum Thurston--Bennequin number for knots

[ Abstract ]

Legendrian submanifolds of contact 3-manifolds are

one-dimensional, just like knots. This ``coincidence'' gives rise to an

interesting and expanding intersection of contact and symplectic geometry

on the one hand and classical knot theory on the other. As an

illustration, we will survey recent results on maximizing the

Thurston--Bennequin number (which is a measure of the twisting of the

contact structure along a Legendrian) within a smooth knot type. In

particular, we will show how Kauffman's state circles can be used to solve

the maximization problem for so-called +adequate (among them, alternating

and positive) knots and links.

Legendrian submanifolds of contact 3-manifolds are

one-dimensional, just like knots. This ``coincidence'' gives rise to an

interesting and expanding intersection of contact and symplectic geometry

on the one hand and classical knot theory on the other. As an

illustration, we will survey recent results on maximizing the

Thurston--Bennequin number (which is a measure of the twisting of the

contact structure along a Legendrian) within a smooth knot type. In

particular, we will show how Kauffman's state circles can be used to solve

the maximization problem for so-called +adequate (among them, alternating

and positive) knots and links.

### 2008/04/22

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

Special fibers of pencils of hypersurfaces

**Sergey Yuzvinsky**(University of Oregon)Special fibers of pencils of hypersurfaces

[ Abstract ]

We consider pencils of hypersurfaces of degree $d>1$ in the complex

$n$-dimensional projective space subject to the condition that the

generic fiber is irreducible. We study the set of completely reducible

fibers, i.e., the unions of hyperplanes. The first surprinsing result is

that the cardinality of thie set has very strict uniformed upper bound

(not depending on $d$ or $n$). The other one gives a characterization

of this set in terms of either topology of its complement or combinatorics

of hyperplanes. We also include into consideration more general special

fibers are iimportant for characteristic varieties of the hyperplane

complements.

We consider pencils of hypersurfaces of degree $d>1$ in the complex

$n$-dimensional projective space subject to the condition that the

generic fiber is irreducible. We study the set of completely reducible

fibers, i.e., the unions of hyperplanes. The first surprinsing result is

that the cardinality of thie set has very strict uniformed upper bound

(not depending on $d$ or $n$). The other one gives a characterization

of this set in terms of either topology of its complement or combinatorics

of hyperplanes. We also include into consideration more general special

fibers are iimportant for characteristic varieties of the hyperplane

complements.

### 2008/04/15

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

On the invariants of knots and 3-manifolds related to the restricted quantum group

**村上 順**(早稲田大学理工)On the invariants of knots and 3-manifolds related to the restricted quantum group

[ Abstract ]

I would like to talk about the colored Alexander invariant and the logarithmic

invariant of knots and links. They are constructed from the universal R-matrices

of the semi-resetricted and restricted quantum groups of sl(2) respectively,

and they are related to the hyperbolic volumes of the cone manifolds along

the knot. I also would like to explain an attempt to generalize these invariants to

a three manifold invariant which relates to the volume of the manifold actually.

I would like to talk about the colored Alexander invariant and the logarithmic

invariant of knots and links. They are constructed from the universal R-matrices

of the semi-resetricted and restricted quantum groups of sl(2) respectively,

and they are related to the hyperbolic volumes of the cone manifolds along

the knot. I also would like to explain an attempt to generalize these invariants to

a three manifold invariant which relates to the volume of the manifold actually.

### 2008/01/29

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

The rotation number function on groups of circle diffeomorphisms

A Diagrammatic Construction of Third Homology Classes of Knot Quandles

**松田 能文**(東京大学大学院数理科学研究科) 16:30-17:30The rotation number function on groups of circle diffeomorphisms

[ Abstract ]

ポアンカレは、円周の向きを保つ同相写像に対して、回転数の有理性と有限軌道の存在が

同値であることを示した。この講演では、この事実が円周の向きを保つ同相写像のなすあ

る種の群に対して一般化できることを説明する。特に、円周の向きを保つ実解析的微分同

相のなす非離散的な群に対して、回転数関数による像の有限性と有限軌道の存在が同値で

あることを示す。

ポアンカレは、円周の向きを保つ同相写像に対して、回転数の有理性と有限軌道の存在が

同値であることを示した。この講演では、この事実が円周の向きを保つ同相写像のなすあ

る種の群に対して一般化できることを説明する。特に、円周の向きを保つ実解析的微分同

相のなす非離散的な群に対して、回転数関数による像の有限性と有限軌道の存在が同値で

あることを示す。

**木村 康人**(東京大学大学院数理科学研究科) 17:30-18:30A Diagrammatic Construction of Third Homology Classes of Knot Quandles

[ Abstract ]

There exists a family of third (quandle / rack) homology classes,

called the shadow (fundamental / diagram) classes,

of the knot quandle, which are obtained from

the shadow colourings of knot diagrams.

We will show the construction of these homology classes,

and also show their relation to the shadow quandle cocycle

invariants of knots and that to other third homology classes.

There exists a family of third (quandle / rack) homology classes,

called the shadow (fundamental / diagram) classes,

of the knot quandle, which are obtained from

the shadow colourings of knot diagrams.

We will show the construction of these homology classes,

and also show their relation to the shadow quandle cocycle

invariants of knots and that to other third homology classes.

### 2008/01/15

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

Adiabatic limits of eta-invariants and the Meyer functions

**飯田 修一**(東京大学大学院数理科学研究科)Adiabatic limits of eta-invariants and the Meyer functions

[ Abstract ]

The Meyer function is the function defined on the hyperelliptic

mapping class group, which gives a signature formula for surface

bundles over surfaces.

In this talk, we introduce certain generalizations of the Meyer

function by using eta-invariants and we discuss the uniqueness of this

function and compute the values for Dehn twists.

The Meyer function is the function defined on the hyperelliptic

mapping class group, which gives a signature formula for surface

bundles over surfaces.

In this talk, we introduce certain generalizations of the Meyer

function by using eta-invariants and we discuss the uniqueness of this

function and compute the values for Dehn twists.

### 2007/12/18

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

Groupoid lifts of representations of mapping classes

**R.C. Penner**(USC and Aarhus University)Groupoid lifts of representations of mapping classes

[ Abstract ]

The "Ptolemy groupoid" is the fundamental path groupoid of the dual to the ideal cell decomposition of the decorated Teichmueller space of a punctured or bordered surface, and the "Torelli groupoid" is thesimilar discretization of the fundamental path groupoid of the quotient

by the Torelli subgroup of mapping classes that acts identically on the first integral homology of the surface. Mapping classes can be represented as appropriate elements of the Ptolemy groupoid and likewise for elements of the Torelli group in the Torelli groupoid.

A natural series of questions is to wonder which representations of mapping class groups, Torelli groups, and their subgroups can be lifted to the groupoid level. In a series of joint works with J. Andersen, A. Bene, N. Kawazumi, and S. Morita, we have given explicit lifts of a number of classical representations: The Johnson representations of the classical and higher Torelli groups

and the symplectic representation of the mapping class group all lift to the Torelli groupoid. Furthermore, the Nielsen representation of the mapping class group as automorphisms of a

free group lifts to the Ptolemy groupoid, and hence so too does any representation

of the mapping class group that factors through its action on the fundamental group of

the surface such as the Magnus representation. We shall survey these various groupoid lifts and discuss current and potential future applications.

The "Ptolemy groupoid" is the fundamental path groupoid of the dual to the ideal cell decomposition of the decorated Teichmueller space of a punctured or bordered surface, and the "Torelli groupoid" is thesimilar discretization of the fundamental path groupoid of the quotient

by the Torelli subgroup of mapping classes that acts identically on the first integral homology of the surface. Mapping classes can be represented as appropriate elements of the Ptolemy groupoid and likewise for elements of the Torelli group in the Torelli groupoid.

A natural series of questions is to wonder which representations of mapping class groups, Torelli groups, and their subgroups can be lifted to the groupoid level. In a series of joint works with J. Andersen, A. Bene, N. Kawazumi, and S. Morita, we have given explicit lifts of a number of classical representations: The Johnson representations of the classical and higher Torelli groups

and the symplectic representation of the mapping class group all lift to the Torelli groupoid. Furthermore, the Nielsen representation of the mapping class group as automorphisms of a

free group lifts to the Ptolemy groupoid, and hence so too does any representation

of the mapping class group that factors through its action on the fundamental group of

the surface such as the Magnus representation. We shall survey these various groupoid lifts and discuss current and potential future applications.

### 2007/12/11

16:30-18:40 Room #056 (Graduate School of Math. Sci. Bldg.)

A Singular Version of The Poincar\\'e-Hopf Theorem

Chen Ruan cohomology of cotangent orbifolds and Chas-Sullivan string topology

**Xavier G\'omez-Mont**(CIMAT, Mexico) 16:30-17:30A Singular Version of The Poincar\\'e-Hopf Theorem

[ Abstract ]

The Poincar\\'e-Hopf Theorem asserts that the Euler Characteristic of a compact manifold is the sum of the indices of any vector field on it with isolated singularities.

A hypersurface in real or complex number space may be considered as the limit of the smooth hypersurfaces obtained from nearby regular values. The singularity contains “hidden” topology, which is unfolded by a smooth regeneration. At the singularity one has an algebraic invariant, the Jacobi Algebra, which is obtained by considering analytic functions modulo the partial derivatives. It contains topological information of the singularity.

One may consider vector fields tangent to a hypersurface with isolated singularities, and define topologically an index, which coincides with the sum of the Poincar\\'e-Hopf indices of a regeneration of it tangent to a nearby smooth hypersurface.

I will explain how to compute the index of a vector field X tangent to an isolated hypersurface singularity V using Homological Algebra, as the Euler Characteristic of the homology of the complex obtained by contracting differential forms on V with the vector field X. The formula contains several terms, but the higher order terms may be translated from the invariants of the singular point to invariants in the Jacobi Algebra, making this translation a local version of the Poincar\\'e-Hopf Theorem.

I will also explain how some of these ideas can be extended to complete intersections.

The Poincar\\'e-Hopf Theorem asserts that the Euler Characteristic of a compact manifold is the sum of the indices of any vector field on it with isolated singularities.

A hypersurface in real or complex number space may be considered as the limit of the smooth hypersurfaces obtained from nearby regular values. The singularity contains “hidden” topology, which is unfolded by a smooth regeneration. At the singularity one has an algebraic invariant, the Jacobi Algebra, which is obtained by considering analytic functions modulo the partial derivatives. It contains topological information of the singularity.

One may consider vector fields tangent to a hypersurface with isolated singularities, and define topologically an index, which coincides with the sum of the Poincar\\'e-Hopf indices of a regeneration of it tangent to a nearby smooth hypersurface.

I will explain how to compute the index of a vector field X tangent to an isolated hypersurface singularity V using Homological Algebra, as the Euler Characteristic of the homology of the complex obtained by contracting differential forms on V with the vector field X. The formula contains several terms, but the higher order terms may be translated from the invariants of the singular point to invariants in the Jacobi Algebra, making this translation a local version of the Poincar\\'e-Hopf Theorem.

I will also explain how some of these ideas can be extended to complete intersections.

**Miguel A. Xicotencatl**(CINVESTAV, Mexico) 17:40-18:40Chen Ruan cohomology of cotangent orbifolds and Chas-Sullivan string topology

[ Abstract ]

(Joint with: A. Gonzalez, E. Lupercio, C. Segovia, and B. Uribe)

At the end of 90's, two theories of topology were invented roughly at the same time and attracted considerable interest in the mathematical community. One is the Chas-Sullivan's loop product on the homology of loop space and the second one is Chen-Ruan's stringy cohomology of orbifold. It was an observation of Chen that inertia orbifold (which carries Chen-Ruan cohomology) is the space of constant loops of an orbifold. Therefore, two theories should interact. In this work we show that for an interesting family of orbifolds, the virtual orbifold cohomology, turns out to be a subalgebra of the homology of the loop orbifold, and is isomorphic, as algebras, to the Chen-Ruan orbifold cohomology of its cotangent orbifold.

(Joint with: A. Gonzalez, E. Lupercio, C. Segovia, and B. Uribe)

At the end of 90's, two theories of topology were invented roughly at the same time and attracted considerable interest in the mathematical community. One is the Chas-Sullivan's loop product on the homology of loop space and the second one is Chen-Ruan's stringy cohomology of orbifold. It was an observation of Chen that inertia orbifold (which carries Chen-Ruan cohomology) is the space of constant loops of an orbifold. Therefore, two theories should interact. In this work we show that for an interesting family of orbifolds, the virtual orbifold cohomology, turns out to be a subalgebra of the homology of the loop orbifold, and is isomorphic, as algebras, to the Chen-Ruan orbifold cohomology of its cotangent orbifold.

### 2007/12/04

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

Morse theory for abelian hyperkahler quotients

**今野 宏**(東京大学大学院数理科学研究科)Morse theory for abelian hyperkahler quotients

[ Abstract ]

In 1980's Kirwan computed Betti numbers of symplectic quotients by using Morse theory. In this talk, we develop this method to hyperkahler quotients by abelian Lie groups. In this method, many computations are much more simplified in the case of hyperkahler quotients than the case of symplectic quotients. As a result we compute not only the Betti numbers, but also the cohomology rings of abelian hyperkahler quotients.

In 1980's Kirwan computed Betti numbers of symplectic quotients by using Morse theory. In this talk, we develop this method to hyperkahler quotients by abelian Lie groups. In this method, many computations are much more simplified in the case of hyperkahler quotients than the case of symplectic quotients. As a result we compute not only the Betti numbers, but also the cohomology rings of abelian hyperkahler quotients.

### 2007/11/27

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

A quandle cocycle invariant for handlebody-links

**石井 敦**(京都大学数理解析研究所)A quandle cocycle invariant for handlebody-links

[ Abstract ]

[joint work with Masahide Iwakiri (Osaka City University)]

A handlebody-link is a disjoint union of circles and a

finite trivalent graph embedded in a closed 3-manifold.

We consider it up to isotopies and IH-moves.

Then it represents an ambient isotopy class of

handlebodies embedded in the closed 3-manifold.

In this talk, I explain how a quandle cocycle invariant

is defined for handlebody-links.

[joint work with Masahide Iwakiri (Osaka City University)]

A handlebody-link is a disjoint union of circles and a

finite trivalent graph embedded in a closed 3-manifold.

We consider it up to isotopies and IH-moves.

Then it represents an ambient isotopy class of

handlebodies embedded in the closed 3-manifold.

In this talk, I explain how a quandle cocycle invariant

is defined for handlebody-links.

### 2007/11/20

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

A certain slice of the character variety of a knot group

and the knot contact homology

**長郷 文和**(東京工業大学大学院理工学研究科)A certain slice of the character variety of a knot group

and the knot contact homology

[ Abstract ]

For a knot $K$ in 3-sphere, we can consider representations of

the knot group $G_K$ into $SL(2,\\mathbb{C})$.

Their characters construct an algebraic set.

This is so-called the $SL(2,\\mathbb{C})$-character variety of

$G_K$ and denoted by $X(G_K)$.

In this talk, we introduce a slice (a subset) $S_0(K)$ of $X(G_K)$.

In fact, this slice is closely related to the A-polynomial

and the abelian knot contact homology.

For example, the A-polynomial $A_K(m,l)$ of a knot $K$ is

a two-variable polynomial knot invariant defined by using

the character variety $X(G_K)$.

Then we can show that for any {\\it small knot} $K$, the number of

irreducible components of $S_0(K)$ gives an upper bound of

the maximal degree of the A-polynomial $A_K(m,l)$ in terms of

the variable $l$.

Moreover, for any 2-bridge knot $K$, we can show that

the coordinate ring of $S_0(K)$ is exactly the degree 0

abelian knot contact homology $HC_0^{ab}(K)$.

We will mainly explain these facts.

For a knot $K$ in 3-sphere, we can consider representations of

the knot group $G_K$ into $SL(2,\\mathbb{C})$.

Their characters construct an algebraic set.

This is so-called the $SL(2,\\mathbb{C})$-character variety of

$G_K$ and denoted by $X(G_K)$.

In this talk, we introduce a slice (a subset) $S_0(K)$ of $X(G_K)$.

In fact, this slice is closely related to the A-polynomial

and the abelian knot contact homology.

For example, the A-polynomial $A_K(m,l)$ of a knot $K$ is

a two-variable polynomial knot invariant defined by using

the character variety $X(G_K)$.

Then we can show that for any {\\it small knot} $K$, the number of

irreducible components of $S_0(K)$ gives an upper bound of

the maximal degree of the A-polynomial $A_K(m,l)$ in terms of

the variable $l$.

Moreover, for any 2-bridge knot $K$, we can show that

the coordinate ring of $S_0(K)$ is exactly the degree 0

abelian knot contact homology $HC_0^{ab}(K)$.

We will mainly explain these facts.

### 2007/11/06

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

Thustion's inequality and open book foliations

**児玉 大樹**(東京大学大学院数理科学研究科)Thustion's inequality and open book foliations

[ Abstract ]

We will study codimension 1 foliations on 3-manifolds.

Thurston's inequality, which implies convexity of the foliation in

some sense, folds for Reebless foliations [Th]. We will discuss

whether the inequality holds or not for open book foliations.

[Th] W. Thurston: Norm on the homology of 3-manifolds, Memoirs of the

AMS, 339 (1986), 99--130.

We will study codimension 1 foliations on 3-manifolds.

Thurston's inequality, which implies convexity of the foliation in

some sense, folds for Reebless foliations [Th]. We will discuss

whether the inequality holds or not for open book foliations.

[Th] W. Thurston: Norm on the homology of 3-manifolds, Memoirs of the

AMS, 339 (1986), 99--130.

### 2007/10/30

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

$L_{\\infty}$ action on Lagrangian filtered $A_{\\infty}$ algebras.

**太田 啓史**(名大多元数理)$L_{\\infty}$ action on Lagrangian filtered $A_{\\infty}$ algebras.

[ Abstract ]

I will discuss $L_{\\infty}$ actions on Lagrangian filtered

$A_{\\infty}$ algebras by cycles of the ambient symplectic

manifold. In the course of the construction,

I like to remark that the stable map compactification is not

sufficient in some case when we consider ones from genus zero

bordered Riemann surface. Also, if I have time, I like to discuss

some relation to (absolute) Gromov-Witten invariant and other

applications.

(This talk is based on my joint work with K.Fukaya, Y-G Oh and K. Ono.)

I will discuss $L_{\\infty}$ actions on Lagrangian filtered

$A_{\\infty}$ algebras by cycles of the ambient symplectic

manifold. In the course of the construction,

I like to remark that the stable map compactification is not

sufficient in some case when we consider ones from genus zero

bordered Riemann surface. Also, if I have time, I like to discuss

some relation to (absolute) Gromov-Witten invariant and other

applications.

(This talk is based on my joint work with K.Fukaya, Y-G Oh and K. Ono.)

### 2007/10/23

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Spaces of subspheres and their applications

**Jun O'Hara**(首都大学東京)Spaces of subspheres and their applications

[ Abstract ]

The set of q-dimensional subspheres in S^n is a Grassmann manifold which has natural pseudo-Riemannian structure, and in some cases, symplectic structure as well. Both of them are conformally invariant.

I will explain some examples of their applications to geometric aspects of knots and links.

The set of q-dimensional subspheres in S^n is a Grassmann manifold which has natural pseudo-Riemannian structure, and in some cases, symplectic structure as well. Both of them are conformally invariant.

I will explain some examples of their applications to geometric aspects of knots and links.

### 2007/10/16

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

Toric Sasaki-Einstein manifolds

**二木 昭人**(東京工業大学大学院理工学研究科)Toric Sasaki-Einstein manifolds

[ Abstract ]

A compact toric Sasaki manifold admits a Sasaki-Einstein metric if and only if it is obtained by the Delzant construction from a toric diagram of a constant height. As an application we see that the canonical line bundle of a toric Fano manifold admits a complete Ricci-flat K\\"ahler metric.

A compact toric Sasaki manifold admits a Sasaki-Einstein metric if and only if it is obtained by the Delzant construction from a toric diagram of a constant height. As an application we see that the canonical line bundle of a toric Fano manifold admits a complete Ricci-flat K\\"ahler metric.

### 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.