## 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

### 2006/10/10

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Elmar Vogt (Frie Universitat Berlin)
Estimating Lusternik-Schnirelmann Category for Foliations:A Survey of Available Techniques
[ Abstract ]
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.

### 2006/07/24

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Boris Hasselblatt (Tufts University)
Invariant foliations in hyperbolic dynamics:
Smoothness and smooth equivalence
[ Abstract ]
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.
[ Reference URL ]
http://faculty.ms.u-tokyo.ac.jp/~topology/

### 2006/07/11

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

[ Abstract ]
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。
[ Reference URL ]
http://faculty.ms.u-tokyo.ac.jp/~topology/

### 2006/07/04

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Alexander A. Ivanov (Imperial College (London))
Amalgams: a machinery of the modern theory of finite groups
[ Reference URL ]
http://faculty.ms.u-tokyo.ac.jp/~topology/

### 2006/06/27

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Cedric Tarquini (Ecole Nomale Superieure of Lyon)
Lorentzian foliations on 3-manifolds
[ Abstract ]
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.

### 2006/06/13

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

A note on C1-moves
[ Abstract ]

### 2006/06/06

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

Thurston's inequality for a foliation with Reeb components
[ Abstract ]
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.

### 2006/05/30

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

[ Abstract ]

### 2006/05/23

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

On crossed homomorphisms on symplectic mapping class groups
[ Abstract ]
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.

### 2006/05/16

17:00-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Laurentiu Maxim (University of Illinois at Chicago)
Fundamental groups of complements to complex hypersurfaces
[ Abstract ]
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.

### 2006/04/25

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

Counting closed orbits and flow lines via Heegaard splittings
[ Abstract ]
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)

### 2006/04/18

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Vladimir Turaev (Univ. Louis Pasteur Strasbourg)
Topology of words
[ Abstract ]
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.

### 2006/04/11

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Martin Arkowitz (Dartmouth College)
Homotopy actions, cyclic maps and their Eckmann-Hilton duals.
[ Abstract ]
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.