## トポロジー火曜セミナー

開催情報 火曜日　17:00～18:30　数理科学研究科棟(駒場) 056号室 河野 俊丈, 河澄 響矢, 北山 貴裕, 逆井卓也 http://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index.html Tea: 16:30 - 17:00 コモンルーム

### 2011年05月17日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

Quandle colorings with non-commutative flows (JAPANESE)
[ 講演概要 ]
This is a joint work with Masahide Iwakiri, Yeonhee Jang and Kanako Oshiro.
We introduce quandle coloring invariants and quandle cocycle invariants
with non-commutative flows for knots, spatial graphs, handlebody-knots,
where a handlebody-knot is a handlebody embedded in the $3$-sphere.
Two handlebody-knots are equivalent if one can be transformed into the
other by an isotopy of $S^3$.
The quandle coloring (resp. cocycle) invariant is a twisted'' quandle
coloring (resp. cocycle) invariant.

### 2011年05月10日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

Isotated points in the space of group left orderings (JAPANESE)
[ 講演概要 ]
The set of all left orderings of a group G admits a natural
topology. In general the space of left orderings is homeomorphic to the
union of Cantor set and finitely many isolated points. In this talk I
will give a new method to construct left orderings corresponding to
isolated points, and will explain how such isolated orderings reflect
the structures of groups.

### 2011年04月26日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム, Lie群論・表現論セミナーと合同

Topological Blow-up (JAPANESE)
[ 講演概要 ]
Suppose that a Lie group $G$ acts on a manifold
$M$. The quotient space $X:=G\\backslash M$ is locally compact,
but not Hausdorff in general. Our aim is to understand
such a non-Hausdorff space $X$.
The space $X$ has the crack $S$. Roughly speaking, $S$ is
the causal subset of non-Hausdorffness of $X$, and especially
$X\\setminus S$ is Hausdorff.

We introduce the concept of topological blow-up' as a repair'
of the crack. The `repaired' space $\\tilde{X}$ is
locally compact and Hausdorff space containing $X\\setminus S$
as its open subset. Moreover, the original space $X$ can be
recovered from the pair of $(\\tilde{X}, S)$.

### 2011年04月12日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

On diffeomorphisms over non-orientable surfaces embedded in the 4-sphere (JAPANESE)
[ 講演概要 ]
4次元球面内に標準的に埋め込まれた向き付け可能曲面上の

### 2011年01月25日(火)

16:30-17:30   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

シート数が小さい曲面結び目の自明化について (JAPANESE)
[ 講演概要 ]
A connected surface smoothly embedded in ${\\mathbb R}^4$ is called a surface-knot. In particular, if a surface-knot $F$ is homeomorphic to the $2$-sphere or the torus, then it is called an $S^2$-knot or a $T^2$-knot, respectively. The sheet number of a surface-knot is an invariant analogous to the crossing number of a $1$-knot. M. Saito and S. Satoh proved some results concerning the sheet number of an $S^2$-knot. In particular, it is known that an $S^2$-knot is trivial if and only if its sheet number is $1$, and there is no $S^2$-knot whose sheet number is $2$. In this talk, we show that there is no $S^2$-knot whose sheet number is $3$, and a $T^2$-knot is trivial if and only if its sheet number is $1$.

### 2011年01月11日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

The Chas-Sullivan conjecture for a surface of infinite genus (JAPANESE)
[ 講演概要 ]

\\Sigma_{\\infty,1} を境界成分 1 の向きづけられたコンパクト曲面の

の中心が定数ループで張られることを証明する。閉曲面についての

することで証明される。時間が許せば、線型コード図式の空間上の
Lie 代数の構造についても議論したい。

### 2010年12月14日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
Kenneth Schackleton 氏 (IPMU)
On the coarse geometry of Teichmueller space (ENGLISH)
[ 講演概要 ]
We discuss the synthetic geometry of the pants graph in
comparison with the Weil-Petersson metric, whose geometry the
pants graph coarsely models following work of Brock’s. We also
restrict our attention to the 5-holed sphere, studying the Gromov
bordification of the pants graph and the dynamics of pseudo-Anosov
mapping classes.

### 2010年12月07日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
Raphael Ponge 氏 (東京大学大学院数理科学研究科)
Diffeomorphism-invariant geometries and noncommutative geometry (ENGLISH)
[ 講演概要 ]
In many geometric situations we may encounter the action of
a group $G$ on a manifold $M$, e.g., in the context of foliations. If
the action is free and proper, then the quotient $M/G$ is a smooth
manifold. However, in general the quotient $M/G$ need not even be
Hausdorff. Furthermore, it is well-known that a manifold has structure
invariant under the full group of diffeomorphisms except the
differentiable structure itself. Under these conditions how can one do
diffeomorphism-invariant geometry?

Noncommutative geometry provides us with the solution of trading the
ill-behaved space $M/G$ for a non-commutative algebra which
essentially plays the role of the algebra of smooth functions on that
space. The local index formula of Atiyah-Singer ultimately holds in
the setting of noncommutative geometry. Using this framework Connes
and Moscovici then obtained in the 90s a striking reformulation of the
local index formula in diffeomorphism-invariant geometry.

An important part the talk will be devoted to reviewing noncommutative
geometry and Connes-Moscovici's index formula. We will then hint to on-
going projects about reformulating the local index formula in two new
geometric settings: biholomorphism-invariant geometry of strictly
pseudo-convex domains and contactomorphism-invariant geometry.

### 2010年11月30日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

Pin^-(2)-monopole equations and intersection forms with local coefficients of 4-manifolds (JAPANESE)
[ 講演概要 ]
We introduce a variant of the Seiberg-Witten equations, Pin^-(2)-monopole equations, and explain its applications to intersection forms with local coefficients of 4-manifolds.
The first application is an analogue of Froyshov's results on 4-manifolds which have definite forms with local coefficients.
The second one is a local coefficient version of Furuta's 10/8-inequality.
As a corollary, we construct nonsmoothable spin 4-manifolds satisfying Rohlin's theorem and the 10/8-inequality.

### 2010年11月16日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

On a colored Khovanov bicomplex (JAPANESE)
[ 講演概要 ]
Jones 多項式の Khovanov ホモロジーと関連理論が近年活発に

Khovanov により対応するコホモロジーが導入され,特に Mackaay と Turner
や Beliakova とWehrli の研究を通し発展した.しかし,このコホモロジーが持つ
2つの境界作用素によって,Khovanov型の複体で2重複体となるものが構成
できるのかは問題として残されていた.もしあるならば Khovanov 型のホモロジーが
Total complexのコホモロジーに収束するスペクトル系列の第2項として理解される.
この問題意識は Beliakova と Wehliの論文によって紹介された.今回はそれに

### 2010年11月09日(火)

17:00-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:40 - 17:00 コモンルーム

Characterising bumping points on deformation spaces of Kleinian groups (JAPANESE)
[ 講演概要 ]
Klein群の変形空間はその内部の相異なる成分がbump,あるいは同一成分が bumpすることがあることが知られている.
Anderson-Canary-McCulloughの研究により,いかなる成分がbumpするかはわかっている.

### 2010年11月02日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
Daniel Ruberman 氏 (Brandeis University)
Periodic-end manifolds and SW theory (ENGLISH)
[ 講演概要 ]
We study an extension of Seiberg-Witten invariants to
4-manifolds with the homology of S^1 \\times S^3. This extension has
many potential applications in low-dimensional topology, including the
study of the homology cobordism group. Because b_2^+ =0, the usual
strategy for defining invariants does not work--one cannot disregard
reducible solutions. In fact, the count of solutions can jump in a
family of metrics or perturbations. To remedy this, we define an
index-theoretic counter-term that jumps by the same amount. The
counterterm is the index of the Dirac operator on a manifold with a
periodic end modeled at infinity by the infinite cyclic cover of the
manifold. This is joint work with Tomasz Mrowka and Nikolai Saveliev.

### 2010年10月26日(火)

17:00-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:40 - 17:00 コモンルーム

Quantum fundamental groups and quantum representation varieties for 3-manifolds (JAPANESE)
[ 講演概要 ]
We define a refinement of the fundamental groups of 3-manifolds and
a generalization of representation variety of the fundamental group
of 3-manifolds. We consider the category $H$ whose morphisms are
nonnegative integers, where $n$ corresponds to a genus $n$ handlebody
equipped with an embedding of a disc into the boundary, and whose
morphisms are the isotopy classes of embeddings of handlebodies
compatible with the embeddings of the disc into the boundaries. For
each 3-manifold $M$ with an embedding of a disc into the boundary, we
can construct a contravariant functor from $H$ to the category of
sets, where the object $n$ of $H$ is mapped to the set of isotopy
classes of embedding of the genus $n$ handlebody into $M$, compatible
with the embeddings of the disc into the boundaries. This functor can
be regarded as a refinement of the fundamental group of $M$, and we
call it the quantum fundamental group of $M$. Using this invariant, we
can construct for each co-ribbon Hopf algebra $A$ an invariant of
3-manifolds which may be regarded as (the space of regular functions
on) the representation variety of $M$ with respect to $A$.

### 2010年10月19日(火)

16:30-18:00   数理科学研究科棟(駒場) 002号室
Tea: 16:00 - 16:30 コモンルーム
Jinseok Cho 氏 (早稲田大学)
Optimistic limits of colored Jones invariants (ENGLISH)
[ 講演概要 ]
Yokota made a wonderful theory on the optimistic limit of Kashaev
invariant of a hyperbolic knot
that the limit determines the hyperbolic volume and the Chern-Simons
invariant of the knot.
Especially, his theory enables us to calculate the volume of a knot
combinatorially from its diagram for many cases.

We will briefly discuss Yokota theory, and then move to the optimistic
limit of colored Jones invariant.
We will explain a parallel version of Yokota theory based on the
optimistic limit of colored Jones invariant.
Especially, we will show the optimistic limit of colored Jones
invariant coincides with that of Kashaev invariant modulo 2\\pi^2.
This implies the optimistic limit of colored Jones invariant also
determines the volume and Chern-Simons invariant of the knot, and

This is a joint-work with Jun Murakami of Waseda University.

### 2010年10月12日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
Andrei Pajitnov 氏 (Univ. de Nantes, 東京大学大学院数理科学研究科)
Asymptotics of Morse numbers of finite coverings of manifolds (ENGLISH)
[ 講演概要 ]
Let X be a closed manifold;
denote by m(X) the Morse number of X
(that is, the minimal number of critical
points of a Morse function on X).
Let Y be a finite covering of X of degree d.

In this survey talk we will address the following question
posed by M. Gromov: What are the asymptotic properties
of m(N) as d goes to infinity?

It turns out that for high-dimensional manifolds with
free abelian fundamental group the asymptotics of
the number m(N)/d is directly related to the Novikov homology
of N. We prove this theorem and discuss related results.

### 2010年07月27日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

Quandle homology and complex volume
(Joint work with Yuichi Kabaya) (JAPANESE)
[ 講演概要 ]
For a hyperbolic 3-manifold M, the complex value (Vol(M) + i CS(M)) is called the complex volume of M. Here, Vol(M) denotes the volume of M, and CS(M) the Chern-Simons invariant of M.
In 2004, Neumann defined the extended Bloch group, and showed that there is an element of the extended Bloch group corresponding to a hyperbolic 3-manifold.
He further showed that we can compute the complex volume of the manifold by evaluating the element of the extended Bloch group.
To obtain an element of the extended Bloch group corresponding to a hyperbolic 3-manifold, we have to find an ideal triangulation of the manifold and to solve several equations.
On the other hand, we show that the element of the extended Bloch group corresponding to the exterior of a hyperbolic link is obtained from a quandle shadow coloring.
It means that we can compute the complex volume combinatorially from a link diagram.

### 2010年07月20日(火)

17:00-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:30 - 17:00 コモンルーム

A polynomial invariant of pseudo-Anosov maps (JAPANESE)
[ 講演概要 ]
Thurston-Nielsen classified surface homomorphism into three classes. Among them, the pseudo-Anosov class is the most interesting since there is strong connection to the hyperbolic manifolds. As an invariant, the dilatation number has been known. In this talk, I will introduce a new invariant of pseudo-Anosov maps. It is an integer coefficient polynomial, which contains the dilatation as the largest real root and is often reducible. I will show properties of the polynomials, examples, and some application to knot theory. (This is a joint work with Joan Birman and Peter Brinkmann.)

### 2010年07月13日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
Marion Moore 氏 (University of California, Davis)
High Distance Knots in closed 3-manifolds (ENGLISH)
[ 講演概要 ]
Let M be a closed 3-manifold with a given Heegaard splitting.
We show that after a single stabilization, some core of the
stabilized splitting has arbitrarily high distance with respect
to the splitting surface. This generalizes a result of Minsky,
Moriah, and Schleimer for knots in S^3. We also show that in the
complex of curves, handlebody sets are either coarsely distinct
or identical. We define the coarse mapping class group of a
Heeegaard splitting, and show that if (S,V,W) is a Heegaard
splitting of genus greater than or equal to 2, then the coarse
mapping class group of (S,V,W) is isomorphic to the mapping class
group of (S, V, W). This is joint work with Matt Rathbun.

### 2010年07月06日(火)

17:00-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:40 - 17:00 コモンルーム

On the cohomology of free and twisted loop spaces (JAPANESE)
[ 講演概要 ]
A natural extension of cohomology suspension to a free loop space is
constructed from the evaluation map and is shown to have a good
properties in cohomology calculation. This map is generalized to a
twisted loop space.
As an application, the cohomology of free and twisted loop space of
classifying spaces of compact Lie groups, including certain finite
Chevalley groups is calculated.

### 2010年06月29日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

Non-commutative Reidemeister torsion and Morse-Novikov theory (JAPANESE)
[ 講演概要 ]
For a circle-valued Morse function of a closed oriented manifold, we
show that Reidemeister torsion over a non-commutative formal Laurent
polynomial ring equals the product of a certain non-commutative
Lefschetz-type zeta function and the algebraic torsion of the Novikov
complex over the ring. This gives a generalization of the results of
Hutchings-Lee and Pazhitnov on abelian coefficients. As a consequence we
obtain Morse theoretical and dynamical descriptions of the higher-order
Alexander polynomials.

### 2010年06月15日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

On exceptional surgeries on Montesinos knots
(joint works with In Dae Jong and Shigeru Mizushima) (JAPANESE)
[ 講演概要 ]
I will report recent progresses of the study on exceptional
surgeries on Montesinos knots.
In particular, we will focus on how homological invariants (e.g.
khovanov homology,
knot Floer homology) on knots can be used in the study of Dehn surgery.

### 2010年06月01日(火)

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: 16:30 - 17:00 コモンルーム

On Fatou-Julia decompositions (JAPANESE)
[ 講演概要 ]
We will explain that Fatou-Julia decompositions can be
introduced in a unified manner to several kinds of one-dimensional
complex dynamical systems, which include the action of Kleinian groups,
iteration of holomorphic mappings and complex codimension-one foliations.
In this talk we will restrict ourselves mostly to the cases where the
dynamical systems have a certain compactness, however, we will mention
how to deal with dynamical systems without compactness.

### 2010年05月18日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

On roots of Dehn twists (JAPANESE)
[ 講演概要 ]
Let $t_{c}$ be the Dehn twist about a nonseparating simple closed curve
$c$ in a closed orientable surface. If a mapping class $f$ satisfies
$t_{c}=f^{n}$ in mapping class group, we call $f$ a root of $t_{c}$ of
degree $n$. In 2009, Margalit and Schleimer constructed roots of $t_{c}$.
In this talk, I will explain the data set which determine a root of
$t_{c}$ up to conjugacy. Moreover, I will explain the minimal and the
maximal degree.

### 2010年05月11日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

The logarithms of Dehn twists (JAPANESE)
[ 講演概要 ]
We establish an explicit formula for the action of (non-separating and
separating) Dehn twists on the complete group ring of the fundamental group of a
surface. It generalizes the classical transvection formula on the first homology.
The proof is involved with a homological interpretation of the Goldman
Lie algebra. This talk is based on a jointwork with Yusuke Kuno (Hiroshima U./JSPS).

### 2010年04月27日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

On the complex volume of hyperbolic knots (JAPANESE)
[ 講演概要 ]
In this talk, we give a formula of the volume and the Chern-Simons invariant of hyperbolic knot complements, which is closely related to the volume conjecture of hyperbolic knots.
We also discuss the volumes and the Chern-Simons invariants of closed 3-manifolds
obtained by Dehn surgeries on hyperbolic knots.