## Tuesday Seminar on Topology

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

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

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

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

**Seminar information archive**

### 2016/06/07

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

Topology of holomorphic Lefschetz pencils on the four-torus (JAPANESE)

**Kenta Hayano**(Keio University)Topology of holomorphic Lefschetz pencils on the four-torus (JAPANESE)

[ Abstract ]

In this talk, we will show that two holomorphic Lefschetz pencils on the four-torus are (smoothly) isomorphic if they have the same genus and divisibility. The proof relies on the theory of moduli spaces of polarized abelian surfaces. We will also give vanishing cycles of some holomorphic pencils of the four-torus explicitly. This is joint work with Noriyuki Hamada (The University of Tokyo).

In this talk, we will show that two holomorphic Lefschetz pencils on the four-torus are (smoothly) isomorphic if they have the same genus and divisibility. The proof relies on the theory of moduli spaces of polarized abelian surfaces. We will also give vanishing cycles of some holomorphic pencils of the four-torus explicitly. This is joint work with Noriyuki Hamada (The University of Tokyo).

### 2016/05/31

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

A linking invariant for algebraic curves (ENGLISH)

**Benoît Guerville-Ballé**(Tokyo Gakugei University)A linking invariant for algebraic curves (ENGLISH)

[ Abstract ]

We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. As an application, we show that this invariant distinguishes a new Zariski pair of curves (ie a pair of curves having same combinatorics, yet different topology).

We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. As an application, we show that this invariant distinguishes a new Zariski pair of curves (ie a pair of curves having same combinatorics, yet different topology).

### 2016/05/24

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

Independence of Roseman moves for surface-knot diagrams (JAPANESE)

**Kokoro Tanaka**(Tokyo Gakugei University)Independence of Roseman moves for surface-knot diagrams (JAPANESE)

[ Abstract ]

Roseman moves are seven types of local modifications for surface-knot diagrams in 3-space which generate ambient isotopies of surface-knots in 4-space. In this talk, I will discuss independence among the seven Roseman moves. In particular, I will focus on Roseman moves involving triple points and on those involving branch points. The former is joint work with Kanako Oshiro (Sophia University) and Kengo Kawamura (Osaka City University), and the latter is joint work with Masamichi Takase (Seikei University).

Roseman moves are seven types of local modifications for surface-knot diagrams in 3-space which generate ambient isotopies of surface-knots in 4-space. In this talk, I will discuss independence among the seven Roseman moves. In particular, I will focus on Roseman moves involving triple points and on those involving branch points. The former is joint work with Kanako Oshiro (Sophia University) and Kengo Kawamura (Osaka City University), and the latter is joint work with Masamichi Takase (Seikei University).

### 2016/05/17

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

Some dynamics of random walks on the mapping class groups (JAPANESE)

**Hidetoshi Masai**(The University of Tokyo)Some dynamics of random walks on the mapping class groups (JAPANESE)

[ Abstract ]

The dynamics of random walks on the mapping class groups on closed surfaces of genus >1 will be discussed. We define the topological entropy of random walks. Then we prove that the drift with respect to Thurston or Teichmüller metrics and the Lyapunov exponent all coincide with the topological entropy. This is a "random version" of pseudo-Anosov dynamics observed by Thurston and I will begin this talk by recalling the work of Thurston.

The dynamics of random walks on the mapping class groups on closed surfaces of genus >1 will be discussed. We define the topological entropy of random walks. Then we prove that the drift with respect to Thurston or Teichmüller metrics and the Lyapunov exponent all coincide with the topological entropy. This is a "random version" of pseudo-Anosov dynamics observed by Thurston and I will begin this talk by recalling the work of Thurston.

### 2016/05/10

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

On Milnor's link-homotopy invariants for handlebody-links (JAPANESE)

**Yuka Kotorii**(The University of Tokyo)On Milnor's link-homotopy invariants for handlebody-links (JAPANESE)

[ Abstract ]

A handlebody-link is a disjoint union of handlebodies embedded in $S^3$ and HL-homotopy is an equivalence relation on handlebody-links generated by self-crossing changes. A. Mizusawa and R. Nikkuni classified the set of HL-homotopy classes of 2-component handlebody-links completely using the linking numbers for handlebody-links. In this talk, by using Milnor's link-homotopy invariants, we construct an invariant for handlebody-links and give a bijection between the set of HL-homotopy classes of n-component handlebody-links with some assumption and a quotient of the action of the general linear group on a tensor product of modules. This is joint work with Atsuhiko Mizusawa at Waseda University.

A handlebody-link is a disjoint union of handlebodies embedded in $S^3$ and HL-homotopy is an equivalence relation on handlebody-links generated by self-crossing changes. A. Mizusawa and R. Nikkuni classified the set of HL-homotopy classes of 2-component handlebody-links completely using the linking numbers for handlebody-links. In this talk, by using Milnor's link-homotopy invariants, we construct an invariant for handlebody-links and give a bijection between the set of HL-homotopy classes of n-component handlebody-links with some assumption and a quotient of the action of the general linear group on a tensor product of modules. This is joint work with Atsuhiko Mizusawa at Waseda University.

### 2016/04/26

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

Arithmetic topology on branched covers of 3-manifolds (JAPANESE)

**Jun Ueki**(The University of Tokyo)Arithmetic topology on branched covers of 3-manifolds (JAPANESE)

[ Abstract ]

The analogy between 3-dimensional topology and number theory was first pointed out by Mazur in the 1960s, and it has been studied systematically by Kapranov, Reznikov, Morishita, and others. In their analogies, for example, knots and 3-manifolds correspond to primes and number rings respectively. The study of these analogies is called arithmetic topology now.

In my talk, based on their dictionary of analogies, we study analogues of idelic class field theory, Iwasawa theory, and Galois deformation theory in the context of 3-dimensional topology, and establish various foundational analogies in arithmetic topology.

The analogy between 3-dimensional topology and number theory was first pointed out by Mazur in the 1960s, and it has been studied systematically by Kapranov, Reznikov, Morishita, and others. In their analogies, for example, knots and 3-manifolds correspond to primes and number rings respectively. The study of these analogies is called arithmetic topology now.

In my talk, based on their dictionary of analogies, we study analogues of idelic class field theory, Iwasawa theory, and Galois deformation theory in the context of 3-dimensional topology, and establish various foundational analogies in arithmetic topology.

### 2016/04/19

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

Topological rigidity of finite cyclic group actions on compact surfaces (ENGLISH)

**Błażej Szepietowski**(Gdansk University)Topological rigidity of finite cyclic group actions on compact surfaces (ENGLISH)

[ Abstract ]

Two actions of a group on a surface are called topologically equivalent if they are conjugate by a homeomorphism of the surface. I will describe a method of enumeration (and classification) of topological equivalence classes of actions of a finite group on a compact surface, based on the combinatorial theory of noneuclidean crystallographic groups (NEC groups in short) and a relationship between the outer automorphism group of an NEC group and certain mapping class group. By this method we study topological equivalence of actions of a finite cyclic group on a compact surface, in the situation where the order of the group is large relative to the genus of the surface.

Two actions of a group on a surface are called topologically equivalent if they are conjugate by a homeomorphism of the surface. I will describe a method of enumeration (and classification) of topological equivalence classes of actions of a finite group on a compact surface, based on the combinatorial theory of noneuclidean crystallographic groups (NEC groups in short) and a relationship between the outer automorphism group of an NEC group and certain mapping class group. By this method we study topological equivalence of actions of a finite cyclic group on a compact surface, in the situation where the order of the group is large relative to the genus of the surface.

### 2016/04/12

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

Homotopy theory of differential graded Lie algebras (ENGLISH)

**Aniceto Murillo**(Universidad de Malaga)Homotopy theory of differential graded Lie algebras (ENGLISH)

[ Abstract ]

Having as motivation the Deligne's principle by which every deformation functor is governed by a differential graded Lie algebra, we build a homotopy theory for these algebras which extend the classical Quillen approach and let us model any (non necessarily 1-connected nor path connected) complex. This is joint work with Urtzi Buijs, Yves Félix and Daniel Tanré.

Having as motivation the Deligne's principle by which every deformation functor is governed by a differential graded Lie algebra, we build a homotopy theory for these algebras which extend the classical Quillen approach and let us model any (non necessarily 1-connected nor path connected) complex. This is joint work with Urtzi Buijs, Yves Félix and Daniel Tanré.

### 2016/04/05

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

Torsion invariants and representation varieties for non-positively curved cube complexes (JAPANESE)

**Takahiro Kitayama**(The University of Tokyo)Torsion invariants and representation varieties for non-positively curved cube complexes (JAPANESE)

[ Abstract ]

Applications of torsion invariants and representation varieties have been extensively studied for 3-manifolds. Twisted Alexander polynomials are known to detect the Thurston norm and fiberedness of a 3-manifold. Ideal points of character varieties are known to detect essential surfaces in a 3-manifold in a certain extension of Culler-Shalen theory. In view of cubulation of 3-manifolds one can expect that these results naturally extend to a wider framework and, in particular, the case of virtually special cube complexes. We formulate and discuss such analogous questions for non-positively curved cube complexes.

Applications of torsion invariants and representation varieties have been extensively studied for 3-manifolds. Twisted Alexander polynomials are known to detect the Thurston norm and fiberedness of a 3-manifold. Ideal points of character varieties are known to detect essential surfaces in a 3-manifold in a certain extension of Culler-Shalen theory. In view of cubulation of 3-manifolds one can expect that these results naturally extend to a wider framework and, in particular, the case of virtually special cube complexes. We formulate and discuss such analogous questions for non-positively curved cube complexes.

### 2016/02/16

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

String Topology, Euler Class and TNCZ free loop fibrations (ENGLISH)

**Luc Menichi**(University of Angers)String Topology, Euler Class and TNCZ free loop fibrations (ENGLISH)

[ Abstract ]

Let $M$ be a connected, closed oriented manifold.

Chas and Sullivan have defined a loop product and a loop coproduct on

$H_*(LM;¥mathbb{F})$, the homology of the

free loops on $M$ with coefficients in the field $¥mathbb{F}$.

By studying this loop coproduct, I will show that if the free loop

fibration

$¥Omega M¥buildrel{i}¥over¥hookrightarrow

LM¥buildrel{ev}¥over¥twoheadrightarrow M$

is homologically trivial, i.e. $i^*:H^*(LM;¥mathbb{F})¥twoheadrightarrow

H^*(¥Omega M;¥mathbb{F})$ is onto,

then the Euler characteristic of $M$ is divisible by the characteristic

of the field $¥mathbb{F}$

(or $M$ is a point).

Let $M$ be a connected, closed oriented manifold.

Chas and Sullivan have defined a loop product and a loop coproduct on

$H_*(LM;¥mathbb{F})$, the homology of the

free loops on $M$ with coefficients in the field $¥mathbb{F}$.

By studying this loop coproduct, I will show that if the free loop

fibration

$¥Omega M¥buildrel{i}¥over¥hookrightarrow

LM¥buildrel{ev}¥over¥twoheadrightarrow M$

is homologically trivial, i.e. $i^*:H^*(LM;¥mathbb{F})¥twoheadrightarrow

H^*(¥Omega M;¥mathbb{F})$ is onto,

then the Euler characteristic of $M$ is divisible by the characteristic

of the field $¥mathbb{F}$

(or $M$ is a point).

### 2016/01/19

15:00-16:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Ricci-mean curvature flows in gradient shrinking Ricci solitons (JAPANESE)

**Hikaru Yamamoto**(The University of Tokyo)Ricci-mean curvature flows in gradient shrinking Ricci solitons (JAPANESE)

[ Abstract ]

A Ricci-mean curvature flow is a coupled parabolic PDE system of a mean

curvature flow and a Ricci flow.

In this talk, we consider a Ricci-mean curvature flow in a gradient

shrinking Ricci soliton, and give a generalization of a well-known result

of Huisken which states that if a mean curvature flow in a Euclidean space

develops a singularity of type I, then its parabolic rescaling near the singular

point converges to a self-shrinker.

A Ricci-mean curvature flow is a coupled parabolic PDE system of a mean

curvature flow and a Ricci flow.

In this talk, we consider a Ricci-mean curvature flow in a gradient

shrinking Ricci soliton, and give a generalization of a well-known result

of Huisken which states that if a mean curvature flow in a Euclidean space

develops a singularity of type I, then its parabolic rescaling near the singular

point converges to a self-shrinker.

### 2016/01/12

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

Heavy subsets and non-contractible trajectories (JAPANESE)

On codimension two contact embeddings in the standard spheres (JAPANESE)

**Morimichi Kawasaki**(The University of Tokyo) 16:30-17:30Heavy subsets and non-contractible trajectories (JAPANESE)

[ Abstract ]

For a compact set Y of an open symplectic manifold $(N,¥omega)$ and a free

homotopy class $¥alpha¥in [S^1,N]$, Biran, Polterovich and Salamon

defined the relative symplectic capacity $C_{BPS}(N,Y;¥alpha)$ which

measures the existence of non-contractible 1-periodic trajectories of

Hamiltonian isotopies.

On the hand, Entov and Polterovich defined heaviness for closed subsets

of a symplectic manifold by using spectral invarinats of the Hamiltonian

Floer theory on contractible trajectories.

Heavy subsets are known to be non-displaceable.

In this talk, we prove the finiteness of $C(M,X,¥alpha)$ (i.e. the

existence of non-contractible 1-periodic trajectories under some setting)

by using heaviness.

For a compact set Y of an open symplectic manifold $(N,¥omega)$ and a free

homotopy class $¥alpha¥in [S^1,N]$, Biran, Polterovich and Salamon

defined the relative symplectic capacity $C_{BPS}(N,Y;¥alpha)$ which

measures the existence of non-contractible 1-periodic trajectories of

Hamiltonian isotopies.

On the hand, Entov and Polterovich defined heaviness for closed subsets

of a symplectic manifold by using spectral invarinats of the Hamiltonian

Floer theory on contractible trajectories.

Heavy subsets are known to be non-displaceable.

In this talk, we prove the finiteness of $C(M,X,¥alpha)$ (i.e. the

existence of non-contractible 1-periodic trajectories under some setting)

by using heaviness.

**Ryo Furukawa**(The University of Tokyo) 17:30-18:30On codimension two contact embeddings in the standard spheres (JAPANESE)

[ Abstract ]

In this talk we consider codimension two contact

embedding problem by using higher dimensional braids.

First, we focus on embeddings of contact $3$-manifolds to the standard $

S^5$ and give some results, for example, any contact structure on $S^3$

can embed so that it is smoothly isotopic to the standard embedding.

These are joint work with John Etnyre. Second, we consider the relative

Euler number of codimension two contact submanifolds and its Seifert

hypersurfaces which is a generalization of the self-linking number of

transverse knots in contact $3$-manifolds. We give a way to calculate

the relative Euler number of certain contact submanifolds obtained by

braids and as an application we give examples of embeddings of one

contact manifold which are isotopic as smooth embeddings but not

isotopic as contact embeddings in higher dimension.

In this talk we consider codimension two contact

embedding problem by using higher dimensional braids.

First, we focus on embeddings of contact $3$-manifolds to the standard $

S^5$ and give some results, for example, any contact structure on $S^3$

can embed so that it is smoothly isotopic to the standard embedding.

These are joint work with John Etnyre. Second, we consider the relative

Euler number of codimension two contact submanifolds and its Seifert

hypersurfaces which is a generalization of the self-linking number of

transverse knots in contact $3$-manifolds. We give a way to calculate

the relative Euler number of certain contact submanifolds obtained by

braids and as an application we give examples of embeddings of one

contact manifold which are isotopic as smooth embeddings but not

isotopic as contact embeddings in higher dimension.

### 2015/12/15

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

The Curved Cartan Complex (ENGLISH)

**Constantin Teleman**(University of California, Berkeley)The Curved Cartan Complex (ENGLISH)

[ Abstract ]

The Cartan model computes the equivariant cohomology of a smooth manifold X with

differentiable action of a compact Lie group G, from the invariant polynomial

functions on the Lie algebra with values in differential forms and a deformation

of the de Rham differential. Before extracting invariants, the Cartan differential

does not square to zero and is apparently meaningless. Unrecognised was the fact

that the full complex is a curved algebra, computing the quotient by G of the

algebra of differential forms on X. This generates, for example, a gauged version of

string topology. Another instance of the construction, applied to deformation

quantisation of symplectic manifolds, gives the BRST construction of the symplectic

quotient. Finally, the theory for a X point with an additional quadratic curving

computes the representation category of the compact group G, and this generalises

to the loop group of G and even to real semi-simple groups.

The Cartan model computes the equivariant cohomology of a smooth manifold X with

differentiable action of a compact Lie group G, from the invariant polynomial

functions on the Lie algebra with values in differential forms and a deformation

of the de Rham differential. Before extracting invariants, the Cartan differential

does not square to zero and is apparently meaningless. Unrecognised was the fact

that the full complex is a curved algebra, computing the quotient by G of the

algebra of differential forms on X. This generates, for example, a gauged version of

string topology. Another instance of the construction, applied to deformation

quantisation of symplectic manifolds, gives the BRST construction of the symplectic

quotient. Finally, the theory for a X point with an additional quadratic curving

computes the representation category of the compact group G, and this generalises

to the loop group of G and even to real semi-simple groups.

### 2015/12/08

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

Lens space surgery and Kirby calculus of 4-manifolds (JAPANESE)

**Yuichi Yamada**(The Univ. of Electro-Comm.)Lens space surgery and Kirby calculus of 4-manifolds (JAPANESE)

[ Abstract ]

The problem asking "Which knot yields a lens space by Dehn surgery" is

called "lens space surgery". Berge's list ('90) is believed to be the

complete list, but it is still unproved, even after some progress by

Heegaard Floer Homology.

This problem seems to enter a new aspect: study using 4-manifolds, lens

space surgery from lens spaces, checking hyperbolicity by computer.

In the talk, we review the structure of Berge's list and talk on our

study on pairs of distinct knots but yield same lens spaces, and

4-maniolds constructed from such pairs. This is joint work with Motoo

Tange (Tsukuba University).

The problem asking "Which knot yields a lens space by Dehn surgery" is

called "lens space surgery". Berge's list ('90) is believed to be the

complete list, but it is still unproved, even after some progress by

Heegaard Floer Homology.

This problem seems to enter a new aspect: study using 4-manifolds, lens

space surgery from lens spaces, checking hyperbolicity by computer.

In the talk, we review the structure of Berge's list and talk on our

study on pairs of distinct knots but yield same lens spaces, and

4-maniolds constructed from such pairs. This is joint work with Motoo

Tange (Tsukuba University).

### 2015/12/01

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

Monodromies of splitting families for singular fibers (JAPANESE)

**Takayuki Okuda**(The University of Tokyo)Monodromies of splitting families for singular fibers (JAPANESE)

[ Abstract ]

A degeneration of Riemann surfaces is a family of complex curves

over a disk allowed to have a singular fiber.

A singular fiber may split into several simpler singular fibers

under a deformation family of such families,

which is called a splitting family for the singular fiber.

We are interested in the topology of splitting families.

For the topological types of degenerations of Riemann surfaces,

it is known that there is a good relationship with

the surface mapping classes, via topological monodromy.

In this talk,

we introduce the "topological monodromies of splitting families",

and give a description of those of certain splitting families.

A degeneration of Riemann surfaces is a family of complex curves

over a disk allowed to have a singular fiber.

A singular fiber may split into several simpler singular fibers

under a deformation family of such families,

which is called a splitting family for the singular fiber.

We are interested in the topology of splitting families.

For the topological types of degenerations of Riemann surfaces,

it is known that there is a good relationship with

the surface mapping classes, via topological monodromy.

In this talk,

we introduce the "topological monodromies of splitting families",

and give a description of those of certain splitting families.

### 2015/11/24

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

On the cohomology ring of the handlebody mapping class group of genus two (JAPANESE)

**Masatoshi Sato**(Tokyo Denki University)On the cohomology ring of the handlebody mapping class group of genus two (JAPANESE)

[ Abstract ]

The genus two handlebody mapping class group acts on a tree

constructed by Kramer from the disk complex,

and decomposes into an amalgamated product of two subgroups.

We determine the integral cohomology ring of the genus two handlebody

mapping class group by examining these two subgroups

and the Mayer-Vietoris exact sequence.

Using this result, we estimate the ranks of low dimensional homology

groups of the genus three handlebody mapping class group.

The genus two handlebody mapping class group acts on a tree

constructed by Kramer from the disk complex,

and decomposes into an amalgamated product of two subgroups.

We determine the integral cohomology ring of the genus two handlebody

mapping class group by examining these two subgroups

and the Mayer-Vietoris exact sequence.

Using this result, we estimate the ranks of low dimensional homology

groups of the genus three handlebody mapping class group.

### 2015/11/17

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

Topology of some three-dimensional singularities related to algebraic geometry (ENGLISH)

**Atsuko Katanaga**(Shinshu University)Topology of some three-dimensional singularities related to algebraic geometry (ENGLISH)

[ Abstract ]

In this talk, we deal with hypersurface isolated singularities. First, we will recall

some topological results of singularities. Next, we will sketch the classification of

singularities in algebraic geometry. Finally, we will focus on the three-dimensional

case and discuss some results obtained so far.

In this talk, we deal with hypersurface isolated singularities. First, we will recall

some topological results of singularities. Next, we will sketch the classification of

singularities in algebraic geometry. Finally, we will focus on the three-dimensional

case and discuss some results obtained so far.

### 2015/11/10

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

Topological T-duality for "Real" circle bundle (JAPANESE)

**Kiyonori Gomi**(Shinshu University)Topological T-duality for "Real" circle bundle (JAPANESE)

[ Abstract ]

Topological T-duality originates from T-duality in superstring theory,

and is first studied by Bouwkneght, Evslin and Mathai. The duality

basically consists of two parts: The first part is that, for any pair

of a principal circle bundle with `H-flux', there is another `T-dual'

pair on the same base space. The second part states that the twisted

K-groups of the total spaces of principal circle bundles in duality

are isomorphic under degree shift. This is the most simple topological

T-duality following Bunke and Schick, and there are a number of

generalizations. The generalization I will talk about is a topological

T-duality for "Real" circle bundles, motivated by T-duality in type II

orbifold string theory. In this duality, a variant of Z_2-equivariant

K-theory appears.

Topological T-duality originates from T-duality in superstring theory,

and is first studied by Bouwkneght, Evslin and Mathai. The duality

basically consists of two parts: The first part is that, for any pair

of a principal circle bundle with `H-flux', there is another `T-dual'

pair on the same base space. The second part states that the twisted

K-groups of the total spaces of principal circle bundles in duality

are isomorphic under degree shift. This is the most simple topological

T-duality following Bunke and Schick, and there are a number of

generalizations. The generalization I will talk about is a topological

T-duality for "Real" circle bundles, motivated by T-duality in type II

orbifold string theory. In this duality, a variant of Z_2-equivariant

K-theory appears.

### 2015/10/27

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

Heegaard Floer homology for graphs (JAPANESE)

**Yuanyuan Bao**(The University of Tokyo)Heegaard Floer homology for graphs (JAPANESE)

[ Abstract ]

Ozsváth and Szabó defined the Heegaard Floer homology (HF) for a closed oriented 3-manifold. The definition was then generalized to links embedded in a 3-manifold and the manifolds with boundary (sutured and bordered manifolds). In the case of links, there is a beautiful combinatorial way to rewrite the original definition of HF, which was defined on a Heegaard diagram of the given link, by using grid diagram. For a balanced bipartite graph, we defined its Heegaard diagram and the HF for it. Around the same time, Harvey and O’Donnol defined the combinatorial HF for transverse graphs (see the definition in [arXiv:1506.04785v1]). In this talk, we compare these two methods.

Ozsváth and Szabó defined the Heegaard Floer homology (HF) for a closed oriented 3-manifold. The definition was then generalized to links embedded in a 3-manifold and the manifolds with boundary (sutured and bordered manifolds). In the case of links, there is a beautiful combinatorial way to rewrite the original definition of HF, which was defined on a Heegaard diagram of the given link, by using grid diagram. For a balanced bipartite graph, we defined its Heegaard diagram and the HF for it. Around the same time, Harvey and O’Donnol defined the combinatorial HF for transverse graphs (see the definition in [arXiv:1506.04785v1]). In this talk, we compare these two methods.

### 2015/10/27

15:00-16:30 Room #056 (Graduate School of Math. Sci. Bldg.)

The unfolded Seiberg-Witten-Floer spectrum and its applications

(ENGLISH)

**Jianfeng Lin**(UCLA)The unfolded Seiberg-Witten-Floer spectrum and its applications

(ENGLISH)

[ Abstract ]

Following Furuta's idea of finite dimensional approximation in

the Seiberg-Witten theory, Manolescu defined the Seiberg-Witten-Floer

stable homotopy type for rational homology three-spheres in 2003. In

this talk, I will explain how to construct similar invariants for a

general three-manifold and discuss some applications of these new

invariants. This is a joint work with Tirasan Khandhawit and Hirofumi

Sasahira.

Following Furuta's idea of finite dimensional approximation in

the Seiberg-Witten theory, Manolescu defined the Seiberg-Witten-Floer

stable homotopy type for rational homology three-spheres in 2003. In

this talk, I will explain how to construct similar invariants for a

general three-manifold and discuss some applications of these new

invariants. This is a joint work with Tirasan Khandhawit and Hirofumi

Sasahira.

### 2015/10/20

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

On the existence of stable compact leaves for

transversely holomorphic foliations (ENGLISH)

**Bruno Scardua**(Universidade Federal do Rio de Janeiro)On the existence of stable compact leaves for

transversely holomorphic foliations (ENGLISH)

[ Abstract ]

One of the most important results in the theory of foliations is

the celebrated Local stability theorem of Reeb :

A compact leaf of a foliation having finite holonomy group is

stable, indeed, it admits a fundamental system of invariant

neighborhoods where each leaf is compact with finite holonomy

group. This result, together with the Global stability theorem of Reeb

(for codimension one real foliations), has many important consequences

and motivates several questions in the theory of foliations. In this talk

we show how to prove:

A transversely holomorphic foliation on a compact complex manifold, exhibits a compact stable

leaf if and only if the set of compact leaves is not a zero measure subset of the manifold.

This is a joint work with Cesar Camacho.

One of the most important results in the theory of foliations is

the celebrated Local stability theorem of Reeb :

A compact leaf of a foliation having finite holonomy group is

stable, indeed, it admits a fundamental system of invariant

neighborhoods where each leaf is compact with finite holonomy

group. This result, together with the Global stability theorem of Reeb

(for codimension one real foliations), has many important consequences

and motivates several questions in the theory of foliations. In this talk

we show how to prove:

A transversely holomorphic foliation on a compact complex manifold, exhibits a compact stable

leaf if and only if the set of compact leaves is not a zero measure subset of the manifold.

This is a joint work with Cesar Camacho.

### 2015/10/06

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

Delooping theorem in K-theory (JAPANESE)

**Sho Saito**(Kavli IPMU)Delooping theorem in K-theory (JAPANESE)

[ Abstract ]

There is an important special class of infinite dimensional vector spaces, formed by those called Tate vector spaces. Since their first appearance in Tate’s work on residues of differentials on curves, they have been playing important roles in several different contexts including the study of formal loop spaces and semi-infinite Hodge theory. They have more sophisticated linear algebraic invariant than finite dimensional vector spaces, for instance the dimension of a Tate vector spaces is not a single integer, but a torsor acted upon by the all integers, and the determinant of an automorphism is not a single invertible scalar, but a torsor acted upon by the all invertible scalars. In this talk I will show how a delooping theorem in K-theory provides a clarified perspective on this phenomenon, using the recently developed higher categorical framework of infinity-topoi.

There is an important special class of infinite dimensional vector spaces, formed by those called Tate vector spaces. Since their first appearance in Tate’s work on residues of differentials on curves, they have been playing important roles in several different contexts including the study of formal loop spaces and semi-infinite Hodge theory. They have more sophisticated linear algebraic invariant than finite dimensional vector spaces, for instance the dimension of a Tate vector spaces is not a single integer, but a torsor acted upon by the all integers, and the determinant of an automorphism is not a single invertible scalar, but a torsor acted upon by the all invertible scalars. In this talk I will show how a delooping theorem in K-theory provides a clarified perspective on this phenomenon, using the recently developed higher categorical framework of infinity-topoi.

### 2015/07/21

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

Ribbon concordance and 0-surgeries along knots (JAPANESE)

**Keiji Tagami**(Tokyo Institute of Technology)Ribbon concordance and 0-surgeries along knots (JAPANESE)

[ Abstract ]

Akbulut and Kirby conjectured that two knots with

the same 0-surgery are concordant. Recently, Yasui

gave a counterexample of this conjecture.

In this talk, we introduce a technique to construct

non-ribbon concordant knots with the same 0-surgery.

Moreover, we give a potential counterexample of the

slice-ribbon conjecture. This is a joint work with

Tetsuya Abe (Osaka City University, OCAMI).

Akbulut and Kirby conjectured that two knots with

the same 0-surgery are concordant. Recently, Yasui

gave a counterexample of this conjecture.

In this talk, we introduce a technique to construct

non-ribbon concordant knots with the same 0-surgery.

Moreover, we give a potential counterexample of the

slice-ribbon conjecture. This is a joint work with

Tetsuya Abe (Osaka City University, OCAMI).

### 2015/07/14

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

How homoclinic orbits explain some algebraic relations holding in Novikov rings. (ENGLISH)

**Carlos Moraga Ferrandiz**(The University of Tokyo, JSPS)How homoclinic orbits explain some algebraic relations holding in Novikov rings. (ENGLISH)

[ Abstract ]

Given u, a de-Rham cohomology class of degree 1 of a closed manifold M, we consider the space F_u of (closed) Morse 1-forms in this class. In Morse theory, it is important to equip each α in F_u with a descending pseudo-gradient X. The case u=0 yields usual Morse theory, while u ≠ 0 yields Morse-Novikov theory, which is devoted to the understanding of the space of equipped 1-forms (α,X) with α in F_u.

Here, X is a descending pseudo-gradient, which is said to be adapted to α.

The morphism π1(M) → R induced by u (given by the integral of any α in F_u over a loop of M) determines a set of u-negative loops.

We show that for every u-negative g in π1(M), there exists a co-dimension 1 C∞-stratum Sg of F_u which is naturally co-oriented. The stratum Sg is made of elements (α, X) such that X has exactly one homoclinic orbit L whose homotopy class is g.

The goal of this talk is to show that there exists a co-dimension 1 C∞-stratum Sg (0) of Sg which lies in the closure of Sg^2. This result explains geometrically an easy algebraic relation holding in the Novikov ring associated with u.

We will mention how this study generalizes to produce some non-evident symmetric formulas holding in the Novikov ring.

Given u, a de-Rham cohomology class of degree 1 of a closed manifold M, we consider the space F_u of (closed) Morse 1-forms in this class. In Morse theory, it is important to equip each α in F_u with a descending pseudo-gradient X. The case u=0 yields usual Morse theory, while u ≠ 0 yields Morse-Novikov theory, which is devoted to the understanding of the space of equipped 1-forms (α,X) with α in F_u.

Here, X is a descending pseudo-gradient, which is said to be adapted to α.

The morphism π1(M) → R induced by u (given by the integral of any α in F_u over a loop of M) determines a set of u-negative loops.

We show that for every u-negative g in π1(M), there exists a co-dimension 1 C∞-stratum Sg of F_u which is naturally co-oriented. The stratum Sg is made of elements (α, X) such that X has exactly one homoclinic orbit L whose homotopy class is g.

The goal of this talk is to show that there exists a co-dimension 1 C∞-stratum Sg (0) of Sg which lies in the closure of Sg^2. This result explains geometrically an easy algebraic relation holding in the Novikov ring associated with u.

We will mention how this study generalizes to produce some non-evident symmetric formulas holding in the Novikov ring.

### 2015/07/07

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

Representation varieties detect essential surfaces (JAPANESE)

**Takahiro Kitayama**(Tokyo Institute of Technology)Representation varieties detect essential surfaces (JAPANESE)

[ Abstract ]

Extending Culler-Shalen theory, Hara and I presented a way to construct

certain kinds of branched surfaces (possibly without any branch) in a 3-

manifold from an ideal point of a curve in the SL_n-character variety.

There exists an essential surface in some 3-manifold known to be not

detected in the classical SL_2-theory. We show that every essential

surface in a 3-manifold is given by the ideal point of a line in the SL_

n-character variety for some n. The talk is partially based on joint

works with Stefan Friedl and Matthias Nagel, and also with Takashi Hara.

Extending Culler-Shalen theory, Hara and I presented a way to construct

certain kinds of branched surfaces (possibly without any branch) in a 3-

manifold from an ideal point of a curve in the SL_n-character variety.

There exists an essential surface in some 3-manifold known to be not

detected in the classical SL_2-theory. We show that every essential

surface in a 3-manifold is given by the ideal point of a line in the SL_

n-character variety for some n. The talk is partially based on joint

works with Stefan Friedl and Matthias Nagel, and also with Takashi Hara.