## 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

### 2014/11/25

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

### 2014/11/18

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

### 2014/11/11

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

### 2014/11/04

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

### 2014/10/21

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

### 2014/10/07

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

### 2014/07/22

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

### 2014/07/08

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

### 2014/07/01

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

### 2014/06/24

17:10-18:10   Room #056 (Graduate School of Math. Sci. Bldg.)
Takefumi Nosaka (Faculty of Mathematics, Kyushu University)
On third homologies of quandles and of groups via Inoue-Kabaya map (JAPANESE)
[ Abstract ]

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

### 2014/06/17

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

### 2014/06/10

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

### 2014/06/03

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

### 2014/05/27

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

### 2014/05/20

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

### 2014/05/13

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

### 2014/04/15

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

### 2014/04/08

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

### 2014/01/21

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

### 2014/01/21

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

### 2014/01/14

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

### 2013/12/24

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

### 2013/12/17

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

### 2013/12/10

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

### 2013/12/03

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