## Tuesday Seminar on Topology

Date, time & place Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) KOHNO Toshitake, KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya Tea: 16:30 - 17:00 Common Room

Seminar information archive

### 2013/10/15

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Masamichi Takase (Seikei University)
Desingularizing special generic maps (JAPANESE)
[ Abstract ]
This is a joint work with Osamu Saeki (IMI, Kyushu University).
A special generic map is a generic map which has only definite
fold as its singularities.
We study the condition for a special generic map from a closed
n-manifold to the p-space (n+1>p), to factor through a codimension
one immersion (or an embedding). In particular, for the cases
where p = 1 and 2 we obtain complete results.
Our techniques are related to Smale-Hirsch theory,
topology of the space of immersions, relation between the space
of topological immersions and that of smooth immersions,
sphere eversions, differentiable structures of homotopy spheres,
diffeomorphism group of spheres, free group actions on the sphere, etc.

### 2013/10/08

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Tatsuro Shimizu (The Univesity of Tokyo)
An invariant of rational homology 3-spheres via vector fields. (JAPANESE)
[ Abstract ]
In this talk, we define an invariant of rational homology 3-spheres with
values in a space $\\mathcal A(\\emptyset)$ of Jacobi diagrams by using
vector fields.
The construction of our invariant is a generalization of both that of
the Kontsevich-Kuperberg-Thurston invariant $z^{KKT}$
and that of Fukaya and Watanabe's Morse homotopy invariant $z^{FW}$.
As an application of our invariant, we prove that $z^{KKT}=z^{FW}$ for
integral homology 3-spheres.

### 2013/10/01

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Naoyuki Monden (Tokyo University of Science)
The geography problem of Lefschetz fibrations (JAPANESE)
[ Abstract ]
To consider holomorphic fibrations complex surfaces over complex curves
and Lefschetz fibrations over surfaces is one method for the study of
complex surfaces of general type and symplectic 4-manifods, respectively.
In this talk, by comparing the geography problem of relatively minimal
holomorphic fibrations with that of relatively minimal Lefschetz
fibrations (i.e., the characterization of pairs $(x,y)$ of certain
invariants $x$ and $y$ corresponding to relatively minimal holomorphic
fibrations and relatively minimal Lefschetz fibrations), we observe the
difference between complex surfaces of general type and symplectic
4-manifolds. In particular, we construct Lefschetz fibrations violating
the slope inequality" which holds for any relatively minimal holomorphic
fibrations.

### 2013/07/16

17:10-18:10   Room #056 (Graduate School of Math. Sci. Bldg.)
On new models of real hyperbolic spaces (JAPANESE)
[ Abstract ]
In this talk, I will introduce several new realization of the real hyperbolic spaces, using classical tools. The constructions will involve aspects of convex geometry as well as projective geometry, and they are interesting from the view point of the history of mathematics. This work belongs to a joint project with Athanase Papadopoulos.

### 2013/07/09

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Ryan Budney (University of Victoria)
Smooth 3-manifolds in the 4-sphere (ENGLISH)
[ Abstract ]
Everyone who has studied topology knows the compact 2-manifolds that embed in the 3-sphere. One dimension up, the problem of which smooth 3-manifolds embed in the 4-sphere turns out to be much more involved with a handful of partial answers. I will describe what is known at the present moment.

### 2013/06/25

17:10-18:10   Room #056 (Graduate School of Math. Sci. Bldg.)
Higher-order generalization of Fukaya's Morse homotopy
invariant of 3-manifolds (JAPANESE)
[ Abstract ]
In his article published in 1996, K. Fukaya constructed
a 3-manifold invariant by using Morse homotopy theory. Roughly, his
invariant is defined by considering several Morse functions on a
3-manifold and counting with weights the ways that the theta-graph can
construction to 3-valent graphs with arbitrary number of loops for
integral homology 3-spheres. I will also discuss extension of our method
to 3-manifolds with positive first Betti numbers.

### 2013/06/18

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Kimihiko Motegi (Nihon University)
Left-orderable, non-L-space surgeries on knots (JAPANESE)
[ Abstract ]
A Dehn surgery is said to be left-orderable
if the resulting manifold of the surgery has the left-orderable fundamental group,
and a Dehn surgery is called an L-space surgery
if the resulting manifold of the surgery is an L-space.
We will focus on left-orderable, non-L-space surgeries on knots in the 3-sphere.
Once we have a knot with left-orderable surgeries,
the periodic construction" enables us to provide infinitely many knots with
left-orderable, non-L-space surgeries.
We apply the construction to present infinitely many hyperbolic knots on each
of which every nontrivial surgery is a left-orderable, non-L-space surgery.
This is a joint work with Masakazu Teragaito.

### 2013/06/11

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Takahiro Kitayama (The University of Tokyo)
On an analogue of Culler-Shalen theory for higher-dimensional
representations
(JAPANESE)
[ Abstract ]
Culler and Shalen established a way to construct incompressible surfaces
in a 3-manifold from ideal points of the SL_2-character variety. We
present an analogous theory to construct certain kinds of branched
surfaces from limit points of the SL_n-character variety. Such a
branched surface induces a nontrivial presentation of the fundamental
group as a 2-dimensional complex of groups. This is a joint work with
Takashi Hara (Osaka University).

### 2013/06/04

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Mustafa Korkmaz (Middle East Technical University)
Low-dimensional linear representations of mapping class groups. (ENGLISH)
[ Abstract ]
For a compact connected orientable surface, the mapping class group
of it is defined as the group of isotopy classes of orientation-preserving
self-diffeomorphisms of S which are identity on the boundary. The action
of the mapping class group on the first homology of the surface
gives rise to the classical 2g-dimensional symplectic representation.
The existence of a faithful linear representation of the mapping class
group is still unknown. In my talk, I will show the following three results;
there is no lower dimensional (complex) linear representation,
up to conjugation the symplectic representation is the unique nontrivial representation in dimension 2g, and there is no faithful linear representation
of the mapping class group in dimensions up to 3g-3. I will also discuss a few applications of these theorems, including some algebraic consequences.

### 2013/05/21

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Yuanyuan Bao (The University of Tokyo)
A Heegaard Floer homology for bipartite spatial graphs and its
properties (ENGLISH)
[ Abstract ]
A spatial graph is a smooth embedding of a graph into a given
3-manifold. We can regard a link as a particular spatial graph.
So it is natural to ask whether it is possible to extend the idea
of link Floer homology to define a Heegaard Floer homology for
spatial graphs. In this talk, we discuss some ideas towards this
question. In particular, we define a Heegaard Floer homology for
bipartite spatial graphs and discuss some further observations
have announced a combinatorial Floer homology for spatial graphs by
considering grid diagrams.

### 2013/05/14

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Kenta Hayano (Osaka University)
Vanishing cycles and homotopies of wrinkled fibrations (JAPANESE)
[ Abstract ]
Wrinkled fibrations on closed 4-manifolds are stable
maps to closed surfaces with only indefinite singularities. Such
fibrations and variants of them have been studied for the past few years
to obtain new descriptions of 4-manifolds using mapping class groups.
Vanishing cycles of wrinkled fibrations play a key role in these studies.
In this talk, we will explain how homotopies of wrinkled fibrtions affect
their vanishing cycles. Part of the results in this talk is a joint work
with Stefan Behrens (Max Planck Institute for Mathematics).

### 2013/05/07

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Tetsuya Ito (RIMS, Kyoto University)
Homological intersection in braid group representation and dual
Garside structure (JAPANESE)
[ Abstract ]
One method to construct linear representations of braid groups is to use
an action of braid groups on certain homology of local system coefficient.
Many famous representations, such as Burau or Lawrence-Krammer-Bigelow
representations are constructed in such a way. We show that homological
intersections on such homology groups are closely related to the dual
Garside structure, a remarkable combinatorial structure of braid, and
prove that some representations detects the length of braids in a
surprisingly simple way.
This work is partially joint with Bert Wiest (Univ. Rennes1).

### 2013/04/30

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Francis Sergeraert (L'Institut Fourier, Univ. de Grenoble)
Discrete vector fields and fundamental algebraic topology.
(ENGLISH)
[ Abstract ]
Robin Forman invented the notion of Discrete Vector Field in 1997.
A recent common work with Ana Romero allowed us to discover the notion
of Eilenberg-Zilber discrete vector field. Giving the topologist a
totally new understanding of the fundamental tools of combinatorial
algebraic topology: Eilenberg-Zilber theorem, twisted Eilenberg-Zilber
theorem, Serre and Eilenberg-Moore spectral sequences,
Eilenberg-MacLane correspondence between topological and algebraic
classifying spaces. Gives also new efficient algorithms for Algebraic
Topology, considerably improving our computer program Kenzo, devoted
to Constructive Algebraic Topology. The talk is devoted to an
introduction to discrete vector fields, the very simple definition of
the Eilenberg-Zilber vector field, and how it can be used in various
contexts.

### 2013/04/23

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Andrei Pajitnov (Univ. de Nantes)
Twisted Novikov homology and jump loci in formal and hyperformal spaces (ENGLISH)
[ Abstract ]
Let X be a CW-complex, G its fundamental group, and R a repesentation of G.
Any element of the first cohomology group of X gives rise to an exponential
deformation of R, which can be considered as a curve in the space of
representations. We show that the cohomology of X with local coefficients
corresponding to the generic point of this curve is computable from a spectral
sequence starting from the cohomology of X with R-twisted coefficients. We
compute the differentials of the spectral sequence in terms of Massey products,
and discuss some particular cases arising in Kaehler geometry when the spectral
sequence degenerates. We explain the relation of these invariants and the
twisted Novikov homology. This is a joint work with Toshitake Kohno.

### 2013/04/09

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Hiroyuki Fuji (The University of Tokyo)
Colored HOMFLY Homology and Super-A-polynomial (JAPANESE)
[ Abstract ]
This talk is based on works in collaboration with S. Gukov, M. Stosic,
and P. Sulkowski. We study the colored HOMFLY homology for knots and
its asymptotic behavior. In recent years, the categorification of the
colored HOMFLY polynomial is proposed in term of homological
discussions via spectral sequence and physical discussions via refined
topological string, and these proposals give the same answer
miraculously. In this talk, we consider the asymptotic behavior of the
colored HOMFLY homology \\`a la the generalized volume conjecture, and
discuss the quantum structure of the colored HOMFLY homology for the
complete symmetric representations via the generalized A-polynomial
which we call “super-A-polynomial”.

### 2013/03/19

16:30-18:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Keiko Kawamuro (University of Iowa)
Open book foliation and application to contact topology (ENGLISH)
[ Abstract ]
Open book foliation is a generalization of Birman and Menasco's braid foliation. Any 3-manifold admits open book decompositions. Open book foliation is a singular foliation on an embedded surface, and is define by the intersection of a surface and the pages of the open book decomposition. By Giroux's identification of open books and contact structures one can use open book foliation method to study contact structures. In this talk I define the open book foliation and show some applications to contact topology. This is joint work with Tetsuya Ito (University of British Columbia).

### 2013/02/19

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Eri Hatakenaka (Tokyo University of Agriculture and Technology)
On the ring of Fricke characters of free groups (JAPANESE)
[ Abstract ]
This is a joint work with Takao Satoh (Tokyo University of Science). We study a descending filtration of the ring of Fricke characters of a free group consisting of ideals on which the automorphism group of the free group naturally acts. Then by using it, we define a descending filtration of the automorphism group of a free group, and investigate a relation between it and the Andreadakis-Johnson filtration.

### 2013/01/22

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Jarek Kedra (University of Aberdeen)
On the autonomous metric of the area preserving diffeomorphism
of the two dimensional disc. (ENGLISH)
[ Abstract ]
Let D be the open unit disc in the Euclidean plane and let
G:=Diff(D, area) be the group of smooth compactly supported
area-preserving diffeomorphisms of D. A diffeomorphism is called
autonomous if it is the time one map of the flow of a time independent
vector field. Every diffeomorphism in G is a composition of a number
of autonomous diffeomorphisms. The least amount of such
diffeomorphisms defines a norm on G. In the talk I will investigate
geometric properties of such a norm.

In particular I will construct a bi-Lipschitz embedding of the free
abelian group of arbitrary rank to G. I will also show that the space
of homogeneous quasi-morphisms vanishing on all autonomous
diffeomorphisms in G is infinite dimensional.

This is a joint work with Michael Brandenbursky.

### 2013/01/21

16:30-17:30   Room #002 (Graduate School of Math. Sci. Bldg.)
Naoki Kato (The University of Tokyo
)
Lie foliations transversely modeled on nilpotent Lie
algebras
(JAPANESE)
[ Abstract ]
To each Lie $\\mathfrak{g}$-foliation, there is an associated subalgebra
$\\mathfrak{h}$ of $\\mathfrak{g}$ with the foliation, which is called the
structure Lie algabra. In this talk, we will explain the inverse problem,
that is, which pair $(\\mathfrak{g},\\mathfrak{h})$ can be realized as a
Lie $\\mathfrak{g}$-foliation with the structure Lie algabra $\\mathfrak{h}$, under the assumption that $\\mathfrak{g}$ is nilpotent.

### 2013/01/21

17:30-18:30   Room #002 (Graduate School of Math. Sci. Bldg.)
Tomohiko Ishida (The University of Tokyo)
Quasi-morphisms on the group of area-preserving diffeomorphisms of
the 2-disk
(JAPANESE)
[ Abstract ]
Gambaudo and Ghys constructed linearly independent countably many quasi-
morphisms on the group of area-preserving diffeomorphisms of the 2-disk
from quasi-morphisms on braid groups.
In this talk, we will explain that their construction is injective as a
homomorphism between vector spaces of quasi-morphisms.
If time permits, we introduce an application by Brandenbursky and K\\c{e}
dra.

### 2012/12/11

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Ismar Volic (Wellesley College)
Homotopy-theoretic methods in the study of spaces of knots and links (ENGLISH)
[ Abstract ]
I will survey the ways in which some homotopy-theoretic
methods, manifold calculus of functors main among them, have in recent
years been used for extracting information about the topology of
be featured. I will end with some recent results about spaces of
homotopy string links and in particular about how one can use functor
calculus in combination with configuration space integrals to extract

### 2012/12/04

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Yoshitake Hashimoto (Tokyo City University)
Conformal field theory for C2-cofinite vertex algebras (JAPANESE)
[ Abstract ]
This is a jount work with Akihiro Tsuchiya (Kavli IPMU).
We consider sheaves of covacua and conformal blocks over parameter spaces of n-pointed Riemann surfaces
for a vertex algebra of which the category of modules is not necessarily semi-simple.
We assume the C2-cofiniteness condition for vertex algebras.
We define "tensor product" of two modules over a C2-cofinite vertex algebra.

### 2012/11/27

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Hiraku Nozawa (JSPS-IHES fellow)
On a finite aspect of characteristic classes of foliations (JAPANESE)
[ Abstract ]
Characteristic classes of foliations are not bounded due to Thurston.
In this talk, we will explain finiteness of characteristic classes for
foliations with certain transverse structures (e.g. transverse
conformally flat structure) and its relation to unboundedness and
rigidity of foliations.
(This talk is based on a joint work with Jesús Antonio
Álvarez López at University of Santiago de Compostela,
which is available as arXiv:1205.3375.)

### 2012/11/20

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Kentaro Nagao (Nagoya University)
3 dimensional hyperbolic geometry and cluster algebras (JAPANESE)
[ Abstract ]
The cluster algebra was discovered by Fomin-Zelevinsky in 2000.
Recently, the structures of cluster algebras are recovered in
many areas including the theory of quantum groups, low
dimensional topology, discrete integrable systems, Donaldson-Thomas
theory, and string theory and there is dynamic development in the
research on these subjects. In this talk I introduce a relation between
3 dimensional hyperbolic geometry and cluster algebras motivated
by some duality in string theory.

### 2012/11/13

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Takahiro Kitayama (RIMS, Kyoto University,JSPS PD)
The virtual fibering theorem and sutured manifold hierarchies (JAPANESE)
[ Abstract ]
In 2007 Agol showed that every irreducible 3-manifold whose fundamental
group is nontrivial and virtually residually finite rationally solvable
(RFRS) is virtually fibered. In the proof he used the theory of
least-weight taut normal surfaces introduced and developed by Oertel and
Tollefson-Wang. We give another proof using complexities of sutured
manifolds. This is a joint work with Stefan Friedl (University of
Cologne).