## Tuesday Seminar on Topology

Seminar information archive ～12/05｜Next seminar｜Future seminars 12/06～

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**

### 2014/11/25

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

Quandle knot invariants and applications (JAPANESE)

**Masahico Saito**(University of South Florida)Quandle knot invariants and applications (JAPANESE)

[ Abstract ]

A quandles is an algebraic structure closely related to knots. Homology theories of

quandles have been defined, and their cocycles are used to construct invariants for

classical knots, spatial graphs and knotted surfaces. In this talk, an overview is given

for quandle cocycle invariants and their applications to geometric properties of knots.

The current status of computations, recent developments and open problems will also

be discussed.

A quandles is an algebraic structure closely related to knots. Homology theories of

quandles have been defined, and their cocycles are used to construct invariants for

classical knots, spatial graphs and knotted surfaces. In this talk, an overview is given

for quandle cocycle invariants and their applications to geometric properties of knots.

The current status of computations, recent developments and open problems will also

be discussed.

### 2014/11/18

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

A Modular Operad of Embedded Curves (ENGLISH)

**Charles Siegel**(Kavli IPMU)A Modular Operad of Embedded Curves (ENGLISH)

[ Abstract ]

Modular operads were introduced by Getzler and Kapranov to formalize the structure of gluing maps between moduli of stable marked curves. We present a construction of analogous gluing maps between moduli of pluri-log-canonically embedded marked curves, which fit together to give a modular operad of embedded curves. This is joint work with Satoshi Kondo and Jesse Wolfson.

Modular operads were introduced by Getzler and Kapranov to formalize the structure of gluing maps between moduli of stable marked curves. We present a construction of analogous gluing maps between moduli of pluri-log-canonically embedded marked curves, which fit together to give a modular operad of embedded curves. This is joint work with Satoshi Kondo and Jesse Wolfson.

### 2014/11/11

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

Unifying unexpected exceptional Dehn surgeries (ENGLISH)

**Kenneth Baker**(University of Miami)Unifying unexpected exceptional Dehn surgeries (ENGLISH)

[ Abstract ]

This past summer Dunfield-Hoffman-Licata produced examples of asymmetric, hyperbolic, 1-cusped 3-manifolds with pairs of lens space Dehn fillings through a search of the extended SnapPea census.

Examinations of these examples with Hoffman and Licata lead us to coincidences with other work in progress that gives a simple holistic topological approach towards producing and extending many of these families. In this talk we'll explicitly describe our construction and discuss related applications of the technique.

This past summer Dunfield-Hoffman-Licata produced examples of asymmetric, hyperbolic, 1-cusped 3-manifolds with pairs of lens space Dehn fillings through a search of the extended SnapPea census.

Examinations of these examples with Hoffman and Licata lead us to coincidences with other work in progress that gives a simple holistic topological approach towards producing and extending many of these families. In this talk we'll explicitly describe our construction and discuss related applications of the technique.

### 2014/11/04

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

The coarse geometry of Teichmuller space. (ENGLISH)

**Brian Bowditch**(University of Warwick)The coarse geometry of Teichmuller space. (ENGLISH)

[ Abstract ]

We describe some results regarding the coarse geometry of the

Teichmuller space

of a compact surface. In particular, we describe when the Teichmuller

space admits quasi-isometric embeddings of euclidean spaces and

half-spaces.

We also give some partial results regarding the quasi-isometric rigidity

of Teichmuller space. These results are based on the fact that Teichmuller

space admits a ternary operation, natural up to bounded distance

which endows it with the structure of a coarse median space.

We describe some results regarding the coarse geometry of the

Teichmuller space

of a compact surface. In particular, we describe when the Teichmuller

space admits quasi-isometric embeddings of euclidean spaces and

half-spaces.

We also give some partial results regarding the quasi-isometric rigidity

of Teichmuller space. These results are based on the fact that Teichmuller

space admits a ternary operation, natural up to bounded distance

which endows it with the structure of a coarse median space.

### 2014/10/21

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

Vanishing theorems for p-local homology of Coxeter groups and their alternating subgroups (JAPANESE)

**Toshiyuki Akita**(Hokkaido University)Vanishing theorems for p-local homology of Coxeter groups and their alternating subgroups (JAPANESE)

[ Abstract ]

Given a prime number $p$, we estimate vanishing ranges of $p$-local homology groups of Coxeter groups (of possibly infinite order) and alternating subgroups of finite reflection groups. Our results generalize those by Nakaoka for symmetric groups and Kleshchev-Nakano and Burichenko for alternating groups. The key ingredient is the equivariant homology of Coxeter complexes.

Given a prime number $p$, we estimate vanishing ranges of $p$-local homology groups of Coxeter groups (of possibly infinite order) and alternating subgroups of finite reflection groups. Our results generalize those by Nakaoka for symmetric groups and Kleshchev-Nakano and Burichenko for alternating groups. The key ingredient is the equivariant homology of Coxeter complexes.

### 2014/10/07

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

Transversality problems in string topology and de Rham chains (JAPANESE)

**Kei Irie**(RIMS, Kyoto University)Transversality problems in string topology and de Rham chains (JAPANESE)

[ Abstract ]

The starting point of string topology is the work of Chas-Sullivan, which uncovered the Batalin-Vilkovisky(BV) structure on homology of the free loop space of a manifold.

It is important to define chain level structures beneath the BV structure on homology, however this problem is yet to be settled.

One of difficulties is that, to define intersection products on chain level, we have to address the transversality issue.

In this talk, we introduce a notion of "de Rham chain" to bypass this trouble, and partially realize expected chain level structures.

The starting point of string topology is the work of Chas-Sullivan, which uncovered the Batalin-Vilkovisky(BV) structure on homology of the free loop space of a manifold.

It is important to define chain level structures beneath the BV structure on homology, however this problem is yet to be settled.

One of difficulties is that, to define intersection products on chain level, we have to address the transversality issue.

In this talk, we introduce a notion of "de Rham chain" to bypass this trouble, and partially realize expected chain level structures.

### 2014/07/22

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

The Index Map and Reciprocity Laws for Contou-Carrere Symbols (ENGLISH)

**Jesse Wolfson**(Northwestern University)The Index Map and Reciprocity Laws for Contou-Carrere Symbols (ENGLISH)

[ Abstract ]

In the 1960s, Atiyah and Janich constructed the families index as a natural map from the space of Fredholm operators to the classifying space of topological K-theory, and showed it to be an equivalence. In joint work with Oliver Braunling and Michael Groechenig, we construct an analogous index map in algebraic K-theory. Building on recent work of Sho Saito, we show this provides an analogue of Atiyah and Janich's equivalence. More significantly, the index map allows us to relate the Contou-Carrere symbol, a local analytic invariant of schemes, to algebraic K-theory. Using this, we provide new proofs of reciprocity laws for Contou-Carrere symbols in dimension 1 (first established by Anderson--Pablos Romo) and 2 (established recently by Osipov--Zhu). We extend these reciprocity laws to all dimensions.

In the 1960s, Atiyah and Janich constructed the families index as a natural map from the space of Fredholm operators to the classifying space of topological K-theory, and showed it to be an equivalence. In joint work with Oliver Braunling and Michael Groechenig, we construct an analogous index map in algebraic K-theory. Building on recent work of Sho Saito, we show this provides an analogue of Atiyah and Janich's equivalence. More significantly, the index map allows us to relate the Contou-Carrere symbol, a local analytic invariant of schemes, to algebraic K-theory. Using this, we provide new proofs of reciprocity laws for Contou-Carrere symbols in dimension 1 (first established by Anderson--Pablos Romo) and 2 (established recently by Osipov--Zhu). We extend these reciprocity laws to all dimensions.

### 2014/07/08

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

The Johnson homomorphism and a family of curve graphs (ENGLISH)

**Ingrid Irmer**(JSPS, the University of Tokyo)The Johnson homomorphism and a family of curve graphs (ENGLISH)

[ Abstract ]

Abstract: A family of curve graphs of an oriented surface $S_{g,1}$ will be defined on which there exists a natural orientation, coming from the orientation of subsurfaces. Distances in these graphs represent commutator lengths in $\\pi_{1}(S_{g,1})$. The displacement of vertices in the graphs under the action of the Torelli group is used to give a combinatorial description of the Johnson homomorphism."

Abstract: A family of curve graphs of an oriented surface $S_{g,1}$ will be defined on which there exists a natural orientation, coming from the orientation of subsurfaces. Distances in these graphs represent commutator lengths in $\\pi_{1}(S_{g,1})$. The displacement of vertices in the graphs under the action of the Torelli group is used to give a combinatorial description of the Johnson homomorphism."

### 2014/07/01

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

Singularities of special Lagrangian submanifolds (JAPANESE)

**Yohsuke Imagi**(Kavli IPMU)Singularities of special Lagrangian submanifolds (JAPANESE)

[ Abstract ]

There are interesting invariants defined by "counting" geometric

objects, such as instantons in dimension 4 and pseudo-holomorphic curves

in symplectic manifolds. To do the counting in a sensible way, however,

we have to care about singularities of the geometric objects. Special

Lagrangian submanifolds seem very difficult to "count" as their

singularities may be very complicated. I'll talk about simple

singularities for which we can make an analogy with instantons in

dimension 4 and pseudo-holomorphic curves in symplectic manifolds. To do

it I'll use some techniques from geometric measure theory and Lagrangian

Floer theory, and the Floer-theoretic part is a joint work with Dominic

Joyce and Oliveira dos Santos.

There are interesting invariants defined by "counting" geometric

objects, such as instantons in dimension 4 and pseudo-holomorphic curves

in symplectic manifolds. To do the counting in a sensible way, however,

we have to care about singularities of the geometric objects. Special

Lagrangian submanifolds seem very difficult to "count" as their

singularities may be very complicated. I'll talk about simple

singularities for which we can make an analogy with instantons in

dimension 4 and pseudo-holomorphic curves in symplectic manifolds. To do

it I'll use some techniques from geometric measure theory and Lagrangian

Floer theory, and the Floer-theoretic part is a joint work with Dominic

Joyce and Oliveira dos Santos.

### 2014/06/24

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

On third homologies of quandles and of groups via Inoue-Kabaya map (JAPANESE)

**Takefumi Nosaka**(Faculty of Mathematics, Kyushu University)On third homologies of quandles and of groups via Inoue-Kabaya map (JAPANESE)

[ Abstract ]

本講演では, 群とその群同型の組から定まるカンドルを扱い, 以下の結果を紹介

する. まず, その際Inoue-Kabaya鎖写像が, カンドルホモロジーから群ホモロ

ジーへの写像とし, 定式化される事を見る. 例えば, 有限体上のAlexander

quandleに対し, 望月3-コサイクル全ては, 当写像を通じ, 或る群コホモロジー

から導出され, 殆どがトリプルマッセイ積で解釈できる事をみる. 加えてカンド

ルの普遍中心拡大に対し, Inoue-Kabaya鎖写像が3次において(或る捩れ部分を

除き)同型となる. なお講演内容は当週にある集中講義の聴講を仮定しない.

本講演では, 群とその群同型の組から定まるカンドルを扱い, 以下の結果を紹介

する. まず, その際Inoue-Kabaya鎖写像が, カンドルホモロジーから群ホモロ

ジーへの写像とし, 定式化される事を見る. 例えば, 有限体上のAlexander

quandleに対し, 望月3-コサイクル全ては, 当写像を通じ, 或る群コホモロジー

から導出され, 殆どがトリプルマッセイ積で解釈できる事をみる. 加えてカンド

ルの普遍中心拡大に対し, Inoue-Kabaya鎖写像が3次において(或る捩れ部分を

除き)同型となる. なお講演内容は当週にある集中講義の聴講を仮定しない.

### 2014/06/17

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

Bounded Euler number of actions of 2-orbifold groups on the circle (JAPANESE)

**Yoshifumi Matsuda**(Aoyama Gakuin University)Bounded Euler number of actions of 2-orbifold groups on the circle (JAPANESE)

[ Abstract ]

Burger, Iozzi and Wienhard defined the bounded Euler number for a

continuous action of the fundamental group of a connected oriented

surface of finite type possibly with punctures on the circle. A Milnor-Wood

type inequality involving the bounded Euler number holds and its maximality

characterizes Fuchsian actions up to semiconjugacy. The definition of the

bounded Euler number can be extended to actions of 2-orbifold groups by

considering coverings. A Milnor-Wood type inequality and the characterization

of Fuchsian actions also hold in this case. In this talk, we describe when lifts

of Fuchsian actions of certain 2-orbifold groups, such as the modular group,

are characterized by its bounded Euler number.

Burger, Iozzi and Wienhard defined the bounded Euler number for a

continuous action of the fundamental group of a connected oriented

surface of finite type possibly with punctures on the circle. A Milnor-Wood

type inequality involving the bounded Euler number holds and its maximality

characterizes Fuchsian actions up to semiconjugacy. The definition of the

bounded Euler number can be extended to actions of 2-orbifold groups by

considering coverings. A Milnor-Wood type inequality and the characterization

of Fuchsian actions also hold in this case. In this talk, we describe when lifts

of Fuchsian actions of certain 2-orbifold groups, such as the modular group,

are characterized by its bounded Euler number.

### 2014/06/10

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

On relation between the Milnor's $¥mu$-invariant and HOMFLYPT

polynomial (JAPANESE)

**Yuka Kotorii**(The University of Tokyo)On relation between the Milnor's $¥mu$-invariant and HOMFLYPT

polynomial (JAPANESE)

[ Abstract ]

Milnor introduced a family of invariants for ordered oriented link,

called $¥bar{¥mu}$-invariants. Polyak showed a relation between the $¥

bar{¥mu}$-invariant of length 3 sequence and Conway polynomial.

Moreover, Habegger-Lin showed that Milnor's invariants are invariants of

string link, called $¥mu$-invariants. We show that any $¥mu$-invariant

of length $¥leq k$ can be represented as a combination of HOMFLYPT

polynomials if all $¥mu$-invariant of length $¥leq k-2$ vanish.

This result is an extension of Polyak's result.

Milnor introduced a family of invariants for ordered oriented link,

called $¥bar{¥mu}$-invariants. Polyak showed a relation between the $¥

bar{¥mu}$-invariant of length 3 sequence and Conway polynomial.

Moreover, Habegger-Lin showed that Milnor's invariants are invariants of

string link, called $¥mu$-invariants. We show that any $¥mu$-invariant

of length $¥leq k$ can be represented as a combination of HOMFLYPT

polynomials if all $¥mu$-invariant of length $¥leq k-2$ vanish.

This result is an extension of Polyak's result.

### 2014/06/03

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

Vector partition functions and the topology of multiple weight varieties

(JAPANESE)

**Tatsuru Takakura**(Chuo University)Vector partition functions and the topology of multiple weight varieties

(JAPANESE)

[ Abstract ]

A multiple weight variety is a symplectic quotient of a direct product

of several coadjoint orbits of a compact Lie group $G$, with respect to

the diagonal action of the maximal torus. Its geometry and topology are

closely related to the combinatorics concerned with the weight space

decomposition of a tensor product of irreducible representations of $G$.

For example, when considering the Riemann-Roch index, we are naturally

lead to the study of vector partition functions with multiplicities.

In this talk, we discuss some formulas for vector partition functions,

especially a generalization of the formula of Brion-Vergne. Then, by

using

them, we investigate the structure of the cohomology of certain multiple

weight varieties of type $A$ in detail.

A multiple weight variety is a symplectic quotient of a direct product

of several coadjoint orbits of a compact Lie group $G$, with respect to

the diagonal action of the maximal torus. Its geometry and topology are

closely related to the combinatorics concerned with the weight space

decomposition of a tensor product of irreducible representations of $G$.

For example, when considering the Riemann-Roch index, we are naturally

lead to the study of vector partition functions with multiplicities.

In this talk, we discuss some formulas for vector partition functions,

especially a generalization of the formula of Brion-Vergne. Then, by

using

them, we investigate the structure of the cohomology of certain multiple

weight varieties of type $A$ in detail.

### 2014/05/27

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

The Teichmuller space and the stable quasiconformal mapping class group for a Riemann surface of infinite type (JAPANESE)

**Ege Fujikawa**(Chiba University)The Teichmuller space and the stable quasiconformal mapping class group for a Riemann surface of infinite type (JAPANESE)

[ Abstract ]

We explain recent developments of the theory of infinite dimensional Teichmuller space. In particular, we observe the dynamics of the orbits by the action of the stable quasiconformal mapping class group on the Teichmuller space and consider the relationship with the asymptotic Teichmuller space. We also introduce the generalized fixed point theorem and the Nielsen realization theorem. Furthermore, we investigate the moduli space of Riemann surface of infinite type.

We explain recent developments of the theory of infinite dimensional Teichmuller space. In particular, we observe the dynamics of the orbits by the action of the stable quasiconformal mapping class group on the Teichmuller space and consider the relationship with the asymptotic Teichmuller space. We also introduce the generalized fixed point theorem and the Nielsen realization theorem. Furthermore, we investigate the moduli space of Riemann surface of infinite type.

### 2014/05/20

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

An application of torus graphs to characterize torus manifolds

with extended actions (JAPANESE)

**Shintaro Kuroki**(The Univeristy of Tokyo)An application of torus graphs to characterize torus manifolds

with extended actions (JAPANESE)

[ Abstract ]

A torus manifold is a compact, oriented 2n-dimensional T^n-

manifolds with fixed points. This notion is introduced by Hattori and

Masuda as a topological generalization of toric manifolds. For a given

torus manifold, we can define a labelled graph called a torus graph (

this may be regarded as a generalization of some class of GKM graphs).

It is known that the equivariant cohomology ring of some nice class of

torus manifolds can be computed by using a combinatorial data of torus

graphs. In this talk, we study which torus action of torus manifolds can

be extended to a non-abelian compact connected Lie group. To do this, we

introduce root systems of (abstract) torus graphs and characterize

extended actions of torus manifolds. This is a joint work with Mikiya

Masuda.

A torus manifold is a compact, oriented 2n-dimensional T^n-

manifolds with fixed points. This notion is introduced by Hattori and

Masuda as a topological generalization of toric manifolds. For a given

torus manifold, we can define a labelled graph called a torus graph (

this may be regarded as a generalization of some class of GKM graphs).

It is known that the equivariant cohomology ring of some nice class of

torus manifolds can be computed by using a combinatorial data of torus

graphs. In this talk, we study which torus action of torus manifolds can

be extended to a non-abelian compact connected Lie group. To do this, we

introduce root systems of (abstract) torus graphs and characterize

extended actions of torus manifolds. This is a joint work with Mikiya

Masuda.

### 2014/05/13

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

Transverse projective structures of foliations and deformations of the Godbillon-Vey class (JAPANESE)

**Taro Asuke**(The University of Tokyo)Transverse projective structures of foliations and deformations of the Godbillon-Vey class (JAPANESE)

[ Abstract ]

Given a smooth family of foliations, we can define the derivative of the Godbillon-Vey class

with respect to the family. The derivative is known to be represented in terms of the projective

Schwarzians of holonomy maps. In this talk, we will study transverse projective structures

and connections, and show that the derivative is in fact determined by the projective structure

and the family.

Given a smooth family of foliations, we can define the derivative of the Godbillon-Vey class

with respect to the family. The derivative is known to be represented in terms of the projective

Schwarzians of holonomy maps. In this talk, we will study transverse projective structures

and connections, and show that the derivative is in fact determined by the projective structure

and the family.

### 2014/04/15

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

On the rational string operations of classifying spaces and the

Hochschild cohomology (JAPANESE)

**Takahito Naito**(The University of Tokyo)On the rational string operations of classifying spaces and the

Hochschild cohomology (JAPANESE)

[ Abstract ]

Chataur and Menichi initiated the theory of string topology of

classifying spaces.

In particular, the cohomology of the free loop space of a classifying

space is endowed with a product

called the dual loop coproduct. In this talk, I will discuss the

algebraic structure and relate the rational dual loop coproduct to the

cup product on the Hochschild cohomology via the Van den Bergh isomorphism.

Chataur and Menichi initiated the theory of string topology of

classifying spaces.

In particular, the cohomology of the free loop space of a classifying

space is endowed with a product

called the dual loop coproduct. In this talk, I will discuss the

algebraic structure and relate the rational dual loop coproduct to the

cup product on the Hochschild cohomology via the Van den Bergh isomorphism.

### 2014/04/08

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

On the number of commensurable fibrations on a hyperbolic 3-manifold. (JAPANESE)

**Hidetoshi Masai**(The University of Tokyo)On the number of commensurable fibrations on a hyperbolic 3-manifold. (JAPANESE)

[ Abstract ]

By work of Thurston, it is known that if a hyperbolic fibred

$3$-manifold $M$ has Betti number greater than 1, then

$M$ admits infinitely many distinct fibrations.

For any fibration $\\omega$ on a hyperbolic $3$-manifold $M$,

the number of fibrations on $M$ that are commensurable in the sense of

Calegari-Sun-Wang to $\\omega$ is known to be finite.

In this talk, we prove that the number can be arbitrarily large.

By work of Thurston, it is known that if a hyperbolic fibred

$3$-manifold $M$ has Betti number greater than 1, then

$M$ admits infinitely many distinct fibrations.

For any fibration $\\omega$ on a hyperbolic $3$-manifold $M$,

the number of fibrations on $M$ that are commensurable in the sense of

Calegari-Sun-Wang to $\\omega$ is known to be finite.

In this talk, we prove that the number can be arbitrarily large.

### 2014/01/21

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

On contact submanifolds of the odd dimensional Euclidean space (JAPANESE)

**Naohiko Kasuya**(The University of Tokyo)On contact submanifolds of the odd dimensional Euclidean space (JAPANESE)

[ Abstract ]

We prove that the Chern class of a closed contact manifold is an

obstruction for codimension two contact embeddings in the odd

dimensional Euclidean space.

By Gromov's h-principle,

for any closed contact $3$-manifold with trivial first Chern class,

there is a contact structure on $\\mathbb{R}^5$ which admits a contact

embedding.

We prove that the Chern class of a closed contact manifold is an

obstruction for codimension two contact embeddings in the odd

dimensional Euclidean space.

By Gromov's h-principle,

for any closed contact $3$-manifold with trivial first Chern class,

there is a contact structure on $\\mathbb{R}^5$ which admits a contact

embedding.

### 2014/01/21

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

Weak eigenvalues in homoclinic classes: perturbations from saddles

with small angles (ENGLISH)

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

with small angles (ENGLISH)

[ Abstract ]

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

new sufficient condition for creating weak contracting eigenvalues is

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

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

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

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

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

heterodimensional cycles. In particular, this sufficient condition is

satisfied by a substantial class of saddles with homoclinic tangencies.

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

new sufficient condition for creating weak contracting eigenvalues is

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

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

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

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

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

heterodimensional cycles. In particular, this sufficient condition is

satisfied by a substantial class of saddles with homoclinic tangencies.

### 2014/01/14

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

State-integral partition functions on shaped triangulations (ENGLISH)

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

[ Abstract ]

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

generalized TQFT within the combinatorial framework of shaped

triangulations with the tetrahedral weight functions given in

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

quantum dilogarithm. By using simple examples, I will

illustrate the connection of this theory with the hyperbolic

geometry in three dimensions.

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

generalized TQFT within the combinatorial framework of shaped

triangulations with the tetrahedral weight functions given in

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

quantum dilogarithm. By using simple examples, I will

illustrate the connection of this theory with the hyperbolic

geometry in three dimensions.

### 2013/12/24

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

Stable homotopy type for monopole Floer homology (ENGLISH)

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

[ Abstract ]

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

construction of stable homotopy type for monopole Floer homology. The

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

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

ingredients are finite dimensional approximation technique and Conley

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

3-manifolds such as the 3-torus.

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

construction of stable homotopy type for monopole Floer homology. The

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

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

ingredients are finite dimensional approximation technique and Conley

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

3-manifolds such as the 3-torus.

### 2013/12/17

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

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

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

[ Abstract ]

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

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

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

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

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

satisfying certain conditions and is an extended notion of an

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

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

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

surface diagrams are related by a finite sequence of ambient isotopies

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

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

well-defined. Further, we give some applications.

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

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

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

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

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

satisfying certain conditions and is an extended notion of an

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

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

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

surface diagrams are related by a finite sequence of ambient isotopies

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

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

well-defined. Further, we give some applications.

### 2013/12/10

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

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

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

[ Abstract ]

Akbulut and Yasui defined cork, and plug

to produce many exotic pairs.

In this talk, we introduce a plug

with respect to Fintushel-Stern's knot surgery

or more 4-dimensional local moves and

and argue by using Heegaard Fleor theory.

Akbulut and Yasui defined cork, and plug

to produce many exotic pairs.

In this talk, we introduce a plug

with respect to Fintushel-Stern's knot surgery

or more 4-dimensional local moves and

and argue by using Heegaard Fleor theory.

### 2013/12/03

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

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

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

[ Abstract ]

(joint work with A. Kolpakov)

We introduce a simple algorithm which transforms every

four-dimensional cubulation into a cusped finite-volume hyperbolic

four-manifold. Combinatorially distinct cubulations give rise to

topologically distinct manifolds. Using this algorithm we construct

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

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

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

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

geometrically a hyperbolic manifold.

(joint work with A. Kolpakov)

We introduce a simple algorithm which transforms every

four-dimensional cubulation into a cusped finite-volume hyperbolic

four-manifold. Combinatorially distinct cubulations give rise to

topologically distinct manifolds. Using this algorithm we construct

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

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

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

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

geometrically a hyperbolic manifold.