Tuesday Seminar on Topology (October, 2005 -- January, 2006)
[Japanese]
[Past Programs]
16:30 -- 18:00 Graduate School of Mathematical Sciences,
The University of Tokyo
Tea: 16:00 -- 16:30 Common Room
Last updated January 23, 2006
Information :
Toshitake Kohno
Nariya Kawazumi
October 11 -- Room 056, 16:30 -- 18:00
Vincent Blanloeil (IRMA, Univ. Louis Pasteur)
Cobordism of surfaces embedded in S4
Abstract:
In this talk I will introduce cobordism of knots, and recall classical results
of this theory. Then I will explain the results of a join work with O. Saeki,
in which we gave a classification of embedded surfaces in S4 up to cobordism.
As a consequence I will give a new proof of Rohlin's theorem on
3-manifolds embeddings in R5.
October 17 -- Room 056, 16:30 -- 18:00
Andrei Pajitnov (Univ. de Nantes)
Novikov homology, twisted Alexander polynomials,
and Thurston norm
Abstract:
We discuss circle-valued Morse functions on
complements to links in the 3-sphere.
A generalization of Morse theory due to S.P.Novikov
leads to computable numerical estimates of the number
of critical points via the twisted Novikov homology,
introduced in a paper of H.Goda and the author. We
discuss a multi-variable analog of the twisted Novikov
homology and its relations to the multi-variable
twisted Alexander polynomials and the Thurston norm.
October 25 -- Room 056, 16:30 -- 18:00
Daniel Matei (Institute of Mathematics of the Romanian Academy)
The local system homology of pure braids on
surfaces
Abstract: Pure braid groups of surfaces, either
closed or punctured, admit the structure of an
iterated extension by free groups. This structure
allows us to construct resolutions of the pure braid
groups over the integers. These resolutions are then
used to derive information on the rank one local
system homology of the pure braids on surfaces.
November 1 -- Room 056, 16:30 -- 18:00
Masahiko Yoshinaga (RIMS, Kyoto University)
Real hyperplane arrangements and Lefschetz's hyperplane section theorem
Abstract: Lefschetz's hyperplane section theorem (LHS) is a result about
topology of complex algebraic varieties. It has some applications
concerning numerical properties of cell decompositions for varieties.
But little is known about attaching maps. In this talk, I will apply
the LHS to the complements of hyperplane arrangements which are defined
over the real number field, and describe the attaching maps by using
combinatorial structures of chambers.
November 22 -- Room 056, 16:30 -- 18:00
Takashi Tsuboi (University of Tokyo)
On the group of real analytic diffeomorphisms
The group of real analytic diffeomorphisms of a real analytic manifold is a rich
group. It is dense in the group of smooth diffeomorphisms. Herman showed that for
the n dimensional torus, its identity component is a simple group. For multi
circle fibered manifolds and products of spheres, we show that its identity
component is a perfect group.
November 29 -- Room 056, 16:30 -- 18:00
Tatsuya Tsukamoto (Waseda University)
The almost alternating diagrams of the trivial knot
Abstract: Bankwitz characterized an alternating diagram representing the trivial knot.
A non-alternating diagram is called almost alternating if
one crossing change makes the diagram alternating.
We characterize an almost alternaing diagram representing the trivial knot.
As a corollary we determine an unknotting number one alternating knot with
a property that the unknotting operation can be done on its alternating diagram.
December 6 -- Room 056, 17:00 -- 18:00
Takao Yamaguchi (University of Tuskuba)
Local structures of 2 dimensional Alexandrov spaces with
curvature bounded from above
December 13 -- Room 056, 16:30-- 18:00
Andres Navas (Universidad de Chile)
On the best regularity for group actions on the interval
Given a continuous action of a finitely generated group on the interval,
we will investigate what is the best regularity that can be achieved
by performing topological conjugacies. As we will see, all the
problem concentrates between the classes C1 and C2.
December 20 -- Room 056, 16:30-- 18:30
Kazuyuki Kuwako (University of Tokyo)
Dehn surgery creating Klein bottles
We consider the situation that Dehn surgery on a knot in the $3$-sphere
yields Klein bottles.
Fix a knot $K \subset S^{3}$ and let $M$ be a complement.
Assume that Dehn surgered manifolds
$M(r)$ and $M(s)$ both contain Klein bottles.
We discuss the geometric intersection number between such two slopes
$r$ and $s$.
Isao Hasegawa (University of Tokyo)
Chart description of monodromy representations
on a closed surface.
Abstract:
In surface-knot theory, S. Kamada introduced chart description
to visualize a monodromy representation of a $2$-dimensional braid.
We define a chart description for a monodromy representation
on a closed surface valued in $G$.
As an application, we describe chart move equivalences
for Lefschetz fibrations and $(Z_2)^m\rtimes S_m$-monodromies.
January 10 -- Room 056, 16:30-- 18:30
Yuuki Tadokoro (University of Tokyo)
A nontrivial algebraic cycle in the Jacobian variety of the Klein
quartic
Abstract:
B. Harris defined the harmonic volume for a compact Riemann surface of
genus $g\geq 3$, by means of Chen's iterated integrals.
We prove some value of the harmonic volume for the Klein quartic $C$ is
nonzero modulo $\frac{1}{2}\mathbb{Z}$, using special values of the
generalized hypergeometric function.
This result tells us the algebraic cycle $C-C^-$ is not algebraically
equivalent to zero in the Jacobian variety $J(C)$.
Hiroyuki Yashima (University of Tokyo)
The primitive vector field and semi-analytic geometry
of reflection groups
Abstract:
The aim of this seminor is to investigate relations
between geometry of a finite irreducible reflection group
and that of its parabolic subgroups.
Fundamental tools are the discriminant of reflection hyperplanes and
the primitive vector field in the categorical quotient space of
reflection group actions. Both of them play important roles in
the theory of the flat (Frobenius) structure of reflection groups and
construction of a semi-algebraic dual polyhedron, invented by K. Saito.
I show that a connected component of common zero point set of
algebroid functions defined by the discriminant
is equivariant homeomorphic to a real vector space
whose dimension is equal to the rank of a parabolic subgroup.
In particular the case that a parabolic subgroup is of corank 1,
this homeomorphism can be extended to semi-analytic diffeomorphism.
January 17 -- Room 056, 16:30 -- 18:00
Ryan Budney (Max Planck Institute)
Topology of spaces of knots
Abstract: I will describe what is known about the homotopy-type of the
space of smooth embeddings of a j-sphere in an n-sphere, Emb(S^j,S^n), and
a closely related space K_{n,j}, the space of "long embeddings" of R^j in
R^n. I will try to survey the "group like" properties of these spaces:
Haefliger's work on the isotopy classes of embeddings of S^j in S^n as
monoids under the connected sum operation, and the iterated loop-space
structures on certain "framed" analogues of K_{n,j}. There are results
of Goodwillie and Weiss on the connectivity of K_{n,j} and Emb(S^j,S^n),
and some computations of non-trivial homotopy groups of K_{n,1} for n>3
due to Sinha, Scannell, Longoni and Cattaneo. I will also give a
description of the homotopy type of Emb(S^1,S^3) and K_{3,1}. The
homotopy type of K_{3,1} is given recursively. Each component of K_{3,1}
is described as the total space of a fiber bundle where the fiber is a
product of "simpler" components of K_{3,1} and the base space can be one
of various "elementary" spaces such as configuration spaces in R^2, and
products of circles.
January 24 -- Room 056, 16:30-- 18:30
Shin-ichi Saiki (University of Tokyo)
Incompressible surfaces in 4-punctured sphere bundles
Abstract:
Hatcher and Thurston classified the incompressible
surfaces in the 2-bridge knot complements.
And Floyd and Hatcher classified the incompressible
surfaces in the 2-bridge link complements
and in the punctured torus bundles over the circle.
A natural extension of the technique used in these works
is to apply for the classification of the incompressible surfaces
in the 4-punctured sphere bundles over the circle.
Indeed,
Floyd and Hatcher mentioned this as a Remark in their paper.
These works do not consider the connectedness of the surfaces.
In this seminar,
we will give a criterion for
the connectedness and (non-)orientability
of an incompressible surface
of 2-bridge knot complement type
in 4-punctured sphere bundles over the circle,
and give the list
of such surfaces with genus 0 or 1.
Tomoo Yokoyama (University of Tokyo)
Minimal codimension one foliations whose fundamental groups of leaves
are isomorphic