## 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

### 2007年01月23日(火)

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

The twistor correspondence for self-dual Zollfrei metrics
----their singularities and reduction

[ 講演概要 ]
C. LeBrun and L. J. Mason investigated a twistor-type correspondence
between indefinite conformal structures of signature (2,2) with some properties
and totally real embeddings from RP^3 to CP^3.
In this talk, two themes following LeBrun and Mason are explained.

First we consider an additional structure:
the conformal structure is equipped with a null surface foliation
which has some singularity.
We establish a global twistor correspondence for such structures,
and we show that a low dimensional correspondence
between some quotient spaces is induced from this twistor correspondence.

Next we formulate a certain singularity for the conformal structures.
We generalize the formulation of LeBrun and Mason's theorem
admitting the singularity, and we show explicit examples.

On the homology group of $Out(F_n)$
[ 講演概要 ]
Suppose $F_n$ is the free group of rank $n$,
$Out(F_n) = Aut(F_n)/Inn(F_n)$ the outer automorphism group of $F_n$.
We compute $H_*(Out(F_n);\\mathbb{Q})$ for $n\\leq 6$ and conclude
that non-trivial classes in this range are generated
by Morita classes $\\mu_i \\in H_{4i}(Out(F_{2i+2});\\mathbb{Q})$.
Also we compute odd primary part of $H^*(Out(F_4);\\mathbb{Z})$.

### 2007年01月16日(火)

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

An $SO(3)$-version of $2$-torsion instanton invariants
[ 講演概要 ]
We construct invariants for simply connected, non-spin $4$-manifolds using torsion cohomology classes of moduli spaces of ASD connections on $SO(3)$-bundles. The invariants are $SO(3)$-version of Fintushel-Stern's $2$-torsion instanton invariants. We show that this $SO(3)$-torsion invariant of $2CP^2 \\# -CP^2$ is non-trivial, while it is known that any invariants of $2CP^2 \\# - CP^2$ coming from the Seiberg-Witten theory are trivial
since $2CP^2 \\# -CP^2$ has a positive scalar curvature metric.

On the non-acyclic Reidemeister torsion for knots
[ 講演概要 ]
The Reidemeister torsion is an invariant of a CW-complex and a representation of its fundamental group. We consider the Reidemeister torsion for a knot exterior in a homology three sphere and a representation given by the composition of an SL(2, C)- (or SU(2)-) representation of the knot group and the adjoint action to the Lie algebra.
We will see that this invariant is expressed by the differential coefficient of the twisted Alexander invariant of the knot and investigate some properties of the invariant by using this relation.

### 2006年12月19日(火)

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

Poisson structures on the homology of the spaces of knots
[ 講演概要 ]
We study the homological properties of the space $K$ of (framed) long knots in $\\R^n$, $n>3$, in particular its Poisson algebra structures.
We had known two kinds of Poisson structures, both of which are based on the action of little disks operad. One definition is via the action on the space $K$. Another comes from the action of chains of little disks on the Hochschild complex of an operad, which appears as $E^1$-term of certain spectral sequence converging to $H_* (K)$. The main result is that these two Poisson structures are the same.
We compute the first non-trivial example of the Poisson bracket. We show that this gives a first example of the homology class of $K$ which does not directly correspond to any chord diagrams.

On projections of pseudo-ribbon sphere-links
[ 講演概要 ]
Suppose $F$ is an embedded closed surface in $R^4$.
We call $F$ a pseudo-ribbon surface link
if its projection is an immersion of $F$ into $R^3$
whose self-intersection set $\\Gamma(F)$ consists of disjointly embedded circles.
H. Aiso classified pseudo-ribbon sphere-knots ($F$ is a sphere.)
when $\\Gamma(F)$ consists of less than 6 circles.
We classify pseudo-ribbon sphere-links
when $F$ is two spheres and $\\Gamma(F)$ consists of less than 7 circles.

### 2006年12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
Maxim Kazarian 氏 (Steklov Math. Institute)
Thom polynomials for maps of curves with isolated singularities
(joint with S. Lando)
[ 講演概要 ]
Thom (residual) polynomials in characteristic classes are used in
the analysis of geometry of functional spaces. They serve as a
tool in description of classes Poincar\\'e dual to subvarieties of
functions of prescribed types. We give explicit universal
expressions for residual polynomials in spaces of functions on
complex curves having isolated singularities and
multisingularities, in terms of few characteristic classes. These
expressions lead to a partial explicit description of a
stratification of Hurwitz spaces.

### 2006年11月28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

The Yamabe constants of infinite coverings and a positive mass theorem
[ 講演概要 ]
The {\\it Yamabe constant} $Y(M, C)$ of a given closed conformal manifold
$(M, C)$ is defined by the infimum of
the normalized total-scalar-curavarure functional $E$
among all metrics in $C$.
The study of the second variation of this functional $E$ led O.Kobayashi and Schoen
to independently introduce a natural differential-topological invariant $Y(M)$,
which is obtained by taking the supremum of $Y(M, C)$ over the space of all conformal classes.
This invariant $Y(M)$ is called the {\\it Yamabe invariant} of $M$.
For the study of the Yamabe invariant,
the relationship between $Y(M, C)$ and those of its conformal coverings
is important, the case when $Y(M, C)> 0$ particularly.
When $Y(M, C) \\leq 0$, by the uniqueness of unit-volume constant scalar curvature metrics in $C$,
the desired relation is clear.
When $Y(M, C) > 0$, such a uniqueness does not hold.
However, Aubin proved that $Y(M, C)$ is strictly less than
the Yamabe constant of any of its non-trivial {\\it finite} conformal coverings,
called {\\it Aubin's Lemma}.
In this talk, we generalize this lemma to the one for the Yamabe constant of
any $(M_{\\infty}, C_{\\infty})$ of its {\\it infinite} conformal coverings,
under a certain topological condition on the relation between $\\pi_1(M)$ and $\\pi_1(M_{\\infty})$.
For the proof of this, we aslo establish a version of positive mass theorem
for a specific class of asymptotically flat manifolds with singularities.

### 2006年11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

High-codimensional knots spun about manifolds
[ 講演概要 ]
The spinning describes several methods of constructing higher-dimensional knots from lower-dimensional knots.
The original spinning (Emil Artin, 1925) has been generalized in various ways. Using one of the most generalized forms of spinning, called "deform-spinning about a submanifold" (Dennis Roseman, 1989), we analyze in a geometric way Haefliger's smoothly knotted (4k-1)-spheres in the 6k-sphere.

### 2006年11月10日(金)

17:40-19:00   数理科学研究科棟(駒場) 118号室

WRT invariant for Seifert manifolds and modular forms
[ 講演概要 ]
We study the SU(2) Witten-Reshetikhin-Turaev invariant for Seifert manifold. We disuss a relationship with the Eichler integral of half-integral modular form and Ramanujan mock theta functions.

### 2006年10月31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

Unsmoothable group actions on elliptic surfaces
[ 講演概要 ]
Let G be a cyclic group of order 3,5 or 7.
We prove the existence of locally linear G-actions on elliptic surfaces which can not be realized by smooth actions with respect to specific smooth structures.
To prove this, we give constraints on smooth actions by using gauge theory.
In fact, we use a mod p vanishing theorem on Seiberg-Witten invariants, which was originally proved by F.Fang.
We give a geometric alternative proof of this, which enables us to extend the theorem.

### 2006年10月24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
Marco Zunino 氏 (JSPS, University of Tokyo)
A review of crossed G-structures
[ 講演概要 ]
We present the definition of "crossed structures" as introduced by Turaev and others a few years ago. One of the original motivations in the introduction of these structures and of the related notion of a "Homotopy Quantum Field Theory" (HQFT) was the extension of Reshetikhin-Turaev invariants to the case of flat principal bundles on 3-manifolds. We resume both this aspect of the theory and other applications in both algebra and topology and we present our results on the algebraic structures involved.

### 2006年10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
Arnaud Deruelle 氏 (University of Tokyo)
Networking Seifert Fibered Surgeries on Knots (joint work with Katura Miyazaki and Kimihiko Motegi)
[ 講演概要 ]
We define a Seifert Surgery Network which consists of integral Dehn surgeries on knots yielding Seifert fiber spaces;here we allow Seifert fiber space with a fiber of index zero as degenerate cases. Then we establish some fundamental properties of the network. Using the notion of the network, we will clarify relationships among known Seifert surgeries. In particular, we demonstrate that many Seifert surgeries are closely connected to those on torus knots in Seifert Surgery Network. Our study suggests that the network enables us to make a global picture of Seifert surgeries.

### 2006年10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
Elmar Vogt 氏 (Frie Universitat Berlin)
Estimating Lusternik-Schnirelmann Category for Foliations:A Survey of Available Techniques
[ 講演概要 ]
The Lusternik-Schnirelmann category of a space $X$ is the smallest number $r$ such that $X$ can be covered by $r + 1$ open sets which are contractible in $X$.For foliated manifolds there are several notions generalizing this concept, all of them due
to Helen Colman. We are mostly concerned with the concept of tangential Lusternik-Schnirelmann category (tangential LS-category). Here one requires a covering by open sets $U$ with the following property. There is a leafwise homotopy starting with the inclusion of $U$ and ending in a map that throws for each leaf $F$ of the foliation each component of $U \\cap F$ onto a single point. A leafwise homotopy is a homotopy moving points only inside leaves. Rather than presenting the still very few results obtained about the LS category of foliations, we survey techniques, mostly quite elementary, to estimate the tangential LS-category from below and above.

### 2006年07月24日(月)

16:30-17:30   数理科学研究科棟(駒場) 056号室

Boris Hasselblatt 氏 (Tufts University)
Invariant foliations in hyperbolic dynamics:
Smoothness and smooth equivalence
[ 講演概要 ]
The stable and unstable leaves of a hyperbolic dynamical system are smooth and form a continuous foliation. Smoothness of this foliation sometimes constrains the topological type of the foliation, other times restricts at least the smooth equivalence class of the dynamical system, or for geodesic flows, the type of the underlying manifold. I will present a broad introduction as well as recent work, work in progress, and open problems.
[ 参考URL ]
http://faculty.ms.u-tokyo.ac.jp/~topology/

### 2006年07月11日(火)

17:00-18:30   数理科学研究科棟(駒場) 056号室

[ 講演概要 ]
n次元多様体上のn個の余次元1葉層構造の組で、n個の葉層構造の接空間の共通部分が各点で0になるものを全葉層と呼ぶ。3次元の場合においては任意の有向閉多様体上に全葉層が存在することが Hardorpによって示されていた。3次元多様体上の全葉層をなす各々の葉層構造の接平面場は互いにホモトピックでありオイラー類が0になることが容易に分かるが、逆にオイラー類が0の平面場を与えたときそれを実現する全葉層が存在するかという問題が自然に生じる。

また、この結果の応用として双接触構造、すなわち横断的に交わる正と負の接触構造の組の存在問題にも触れたい。
[ 参考URL ]
http://faculty.ms.u-tokyo.ac.jp/~topology/

### 2006年07月04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
Alexander A. Ivanov 氏 (Imperial College (London))
Amalgams: a machinery of the modern theory of finite groups
[ 参考URL ]
http://faculty.ms.u-tokyo.ac.jp/~topology/

### 2006年06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
Cedric Tarquini 氏 (Ecole Nomale Superieure of Lyon)
Lorentzian foliations on 3-manifolds
[ 講演概要 ]
a joint work with C. Boubel (Ecole Nomale Superieure of Lyon) and P. Mounoud (University of Bordeaux 1 sciences and technologies)

The aim of this work is to give a classification of transversely Lorentzian one dimensional foliations on compact manifolds of dimension three. There are the foliations which admit a transverse pseudo-Riemanniann metric of index one. It is the Lorentzian analogue of the better known Riemannian foliations and they still have rigid transverse geometry.

The Riemannian case was listed by Y. Carriere and we will see that the Lorentzian one is very different and much more complicated to classify. The difference comes form the fact that the completness of the transverse structure, which is automatic in the Riemannian case, is a very strong hypothesis for a transverse Lorentzian foliation.

We will give a classification of complete Lorentzian foliations and some examples which are not complete. As a natural corollary of this classification we will list the codimension one timelike geodesically complete totally geodesic foliations of Lorentzian compact three manifolds.

### 2006年06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

A note on C1-moves
[ 講演概要 ]

### 2006年06月06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

Thurston's inequality for a foliation with Reeb components
[ 講演概要 ]
The Euler class of a Reebless foliation or a tight contact structure on a closed 3-manifold satisfies Thurston's inequality, i.e. its (dual) Thurston norm is less than or equal to 1. It should be significant to study Thurston's inequality in both of foliation theory and contact topology. We investigate conditions for a spinnable foliation one of which assures that Thurston's inequality holds and also another of which implies the violation of the inequality.

### 2006年05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

[ 講演概要 ]

### 2006年05月23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 crossed homomorphisms on symplectic mapping class groups
[ 講演概要 ]
We consider a symplectic manifold M. For a relation between Chern classes of M and the cohomology class of the symplectic form, we construct a crossed homomorphism on the symplectomorphism group of M with values in the cohomology group of M. We show an application of it to the flux homomorphism. Then we see that it induces a one on the symplectic mapping class group of M and show a nontrivial example of it.

### 2006年05月16日(火)

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: 16:40 - 17:00 コモンルーム
Laurentiu Maxim 氏 (University of Illinois at Chicago)
Fundamental groups of complements to complex hypersurfaces
[ 講演概要 ]
I will survey various Alexander-type invariants of hypersurface complements, with an emphasis on obstructions on the type of groups that can arise as fundamental groups of complements to affine hypersurfaces.

### 2006年04月25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

Counting closed orbits and flow lines via Heegaard splittings
[ 講演概要 ]
Let K be a fibred knot in the 3-sphere. It is known that the Alexander polynomial of K is essentially equal to a Lefschetz zeta function obtained from the monodromy map of the fibre structure. In this talk, we discuss the non-fibred knot case. We introduce "monodromy matrix" by making use of Heegaard splitting for sutured manifolds of a knot K, and then observe a method of counting closed orbits and flow lines, which gives the Alexander polynomial of K. This observation is based on works of Donaldson and Mark. (joint work with Hiroshi Matsuda and Andrei Pajitnov)

### 2006年04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
Vladimir Turaev 氏 (Univ. Louis Pasteur Strasbourg)
Topology of words
[ 講演概要 ]
There is a parallel between words, defined as finite sequences of letters, and curves on surfaces. This allows to treat words as geometric objects and to analyze them using techniques from low-dimensional topology. I will discuss basic ideas in this direction and the resulting topological invariants of words.

### 2006年04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
Martin Arkowitz 氏 (Dartmouth College)
Homotopy actions, cyclic maps and their Eckmann-Hilton duals.
[ 講演概要 ]
We study the homotopy action of a based space A on a based space X. The resulting map A--->X is called cyclic. We classify actions on an H-space which are compatible with the H-structure. In the dual case we study coactions X--->X v B and the resulting cocyclic map X--->B. We relate the cocyclicity of a map to the Lusternik-Schnirelmann category of the map.