Tuesday Seminar on Topology (October, 2008 -- March, 2009)
[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 February 9, 2009
Information :
Toshitake Kohno
Nariya Kawazumi
October 14 -- Room 056, 16:30 -- 18:00
Jeffrey Herschel Giansiracusa (Oxford University)
Pontrjagin-Thom maps and the Deligne-Mumford compactification
Abstract:
An embedding f: M -> N produces, via a construction of Pontrjagin-Thom, a map from N to the Thom space of the normal bundle over M. If f is an arbitrary map then one instead gets a map from N to the infinite loop space of the Thom spectrum of the normal bundle of f. We extend this Pontrjagin-Thom construction of wrong-way maps to differentiable stacks and use it to produce interesting maps from the Deligne-Mumford compactification of the moduli space of curves to certain infinite loop spaces. We show that these maps are surjective on mod p homology in a range of degrees. We thus produce large new families of torsion cohomology classes on the Deligne-Mumford compactification.
October 21 -- Room 002, 16:30 -- 18:00
Tetsuhiro Moriyama (The University of Tokyo)
On embeddings of 3-manifolds in 6-manifolds
Abstract:
In this talk, we give a simple axiomatic definition of an invariant of
smooth embeddings of 3-manifolds in 6-manifolds.
The axiom is expressed in terms of some cobordisms of pairs of manifolds of
dimensions 6 and 3 (equipped with some cohomology class of the complement) and
the signature of 4-manifolds.
We then show that our invariant gives a unified framework for some classical
invariants in low-dimensions (Haefliger invariant, Milnor's triple
linking number, Rokhlin invariant, Casson invariant,
Takase's invariant, Skopenkov's invariants).
October 28 -- Room 056, 16:30 -- 18:00
Kazuhiko Kiyono (The University of Tokyo)
Nonsmoothable group actions on spin 4-manifolds
Abstract:
We call a locally linear group action on a topological manifold nonsmoothable
if the action is not smooth with respect to any possible smooth structure.
We show in this lecture that every closed, simply connected, spin topological 4-manifold
not homeomorphic to S2 x S2 or S4 allows a nonsmoothable
group action of any cyclic group with sufficiently large prime order
which depends on the manifold.
November 4 -- Room 056, 16:30 -- 18:00
Misha Verbitsky (ITEP, Moscow)
Lefschetz SL(2)-action and cohomology of Kaehler manifolds.
Abstract:
Let M be compact Kaehler manifold. It is well
known that any Kaehler form generates a Lefschetz SL(2)-triple
acting on cohomology of M. This action can be used to compute
cohomology of M. If M is a hyperkaehler manifold, of real
dimension 4n, then the subalgebra of its cohomology generated by
the second cohomology is isomorphic to a polynomial algebra,
up to the middle degree.
November 11 -- Room 056, 16:30 -- 18:00
Thomas Andrew Putman (MIT)
The second rational homology group of the moduli space of curves
with level structures
Abstract:
Let $\Gamma$ be a finite-index subgroup of the mapping class
group of a closed genus $g$ surface that contains the Torelli group. For
instance, $\Gamma$ can be the level $L$ subgroup or the spin mapping class
group. We show that $H_2(\Gamma;\Q) \cong \Q$ for $g \geq 5$. A corollary
of this is that the rational Picard groups of the associated finite covers
of the moduli space of curves are equal to $\Q$. We also prove analogous
results for surface with punctures and boundary components.
November 18 -- Room 056, 17:00 -- 18:00
Mitsuhiro Shishikura (Kyoto University)
Renormalization and regidity of complex dynamical systems
November 25 -- Room 056, 16:30 -- 18:00
Andrei Pajitnov (Université de Nantes)
Circle-valued Morse theory for knots and links
Abstract:
We will discuss several recent developments in
this theory. In the first part of the talk we prove that the Morse-Novikov number of a knot is less than or equal to twice the tunnel number of the knot, and present consequences of this result. In the second part we report on our joint project with Hiroshi Goda on the half-transversal Morse-Novikov theory for 3-manifolds.
December 2 -- Room 056, 17:00 -- 18:00
Masahiko Kanai (Nagoya University)
Vanishing and Rigidity
Abstract: The aim of my talk is to reveal an unforeseen link between
the classical vanishing theorems of Matsushima and Weil, on the one hand,
andrigidity of the Weyl chamber flow, a dynamical system arising from a higher-rank
noncompact Lie group, on the other. The connection is established via
"transverse extension theorems": Roughly speaking, they claim that a tangential 1- form of the
orbit foliation of the Weyl chamber flow that is tangentially closed
(and satisfies a certain mild additional condition) can be extended to a closed 1- form on the
whole space in a canonical manner. In particular, infinitesimal
rigidity of the orbit foliation of the Weyl chamber flow is proved as an application.
December 9 -- Room 056, 16:30 -- 18:00
Bertrand Deroin (CNRS, Orsay, Université Paris-Sud 11)
Tits alternative in Diff1(S1)
Abstract:
The following form of Tits alternative for subgroups of
homeomorphisms of the circle has been proved by Margulis: or the group
preserve a probability measure on the circle, or it contains a free
subgroup on two generators. We will prove that if the group acts by diffeomorphisms of
class C1 and does not preserve a probability measure on the circle, then
in fact it contains a subgroup topologically conjugated to a Schottky group.
This is a joint work with V. Kleptsyn and A. Navas.
January 13 -- Room 002, 16:30 -- 17:30
Atsushi Yamashita (The University of Tokyo)
Compactification of the homeomorphism group of a graph
Abstract: Topological properties of homeomorphism groups, especially of finite-dimensional manifolds,
have been of interest in the area of infinite-dimensional manifold topology.
For a locally finite graph $\Gamma$ with countably many components,
the homeomorphism group $\mathcal{H}(\Gamma)$
and its identity component $\mathcal{H}_+(\Gamma)$ are topological groups
with respect to the compact-open topology. I will define natural compactifications
$\overline{\mathcal{H}}(\Gamma)$ and
$\overline{\mathcal{H}}_+(\Gamma)$ of these groups and describe the
topological type of the pair $(\overline{\mathcal{H}}_+(\Gamma), \mathcal{H}_+(\Gamma))$
using the data of $\Gamma$. I will also discuss the topological structure of
$\overline{\mathcal{H}}(\Gamma)$ where $\Gamma$ is the circle.
January 20 -- Room 002, 16:30 -- 17:30
Hiraku Nozawa (The University of Tokyo)
Five dimensional $K$-contact manifolds of rank 2
Abstract:
A $K$-contact manifold is an odd dimensional manifold $M$ with a contact form $\alpha$ whose Reeb flow preserves a Riemannian metric on $M$. For examples, the underlying manifold with the underlying contact form of a Sasakian manifold is $K$-contact. In this talk, we will state our results on classification up to surgeries, the existence of compatible Sasakian metrics and a sufficient condition to be toric for closed $5$-dimensional $K$-contact manifolds with a $T^2$ action given by the closure of the Reeb flow, which are obtained by the application of Morse theory to the contact moment map for the $T^2$ action.
17:30 -- 18:30
Inasa Nakamura (The University of Tokyo)
Surface links which are coverings of a trivial torus knot
Abstract:
We consider surface links which are in the form of coverings of a
trivial torus knot, which we will call torus-covering-links.
By definition, torus-covering-links include
spun T2-knots, turned spun T2-knots, and symmetry-spun tori.
We see some properties of torus-covering-links.
January 27 -- Room 056, 17:00 -- 18:00
Kenji Fukaya (Kyoto University)
Lagrangian Floer homology and quasi homomorphism
from the group of Hamiltonian diffeomorphism
Abstract:
Entov-Polterovich constructed quasi homomorphism
from the group of Hamiltonian diffeomorphisms using
spectral invariant due to Oh etc.
In this talk I will explain a way to study this
quasi homomorphism by using Lagrangian Floer homology.
I will also explain its generalization to use quantum
cohomology with bulk deformation.
When applied to the case of toric manifold, it
gives an example where (infinitely) many quasi homomorphism
exists.
(Joint work with Oh-Ohta-Ono).
March 5 (Thursday) -- Room 056, 16:30 -- 18:00
Shicheng Wang (Peking University)
Extending surface automorphisms over 4-space
Abstract:
Let $e: M^p\to R^{p+2}$ be a co-dimensional 2 smooth embedding
from a closed orientable manifold to the Euclidean space and $E_e$ be the subgroup of ${\cal M}_M$, the mapping class group
of $M$, whose elements extend over $R^{p+2}$ as self-diffeomorphisms. Then there is a spin structure
on $M$ derived from the embedding which is preserved by each $\tau \in E_e$.
Some applications: (1) the index $[{\cal M}_{F_g}:E_e]\geq 2^{2g-1}+2^{g-1}$ for any embedding $e:F_g\to R^4$, where $F_g$
is the surface of genus $g$. (2) $[{\cal M}_{T^p}:E_e]\geq 2^p-1$ for any unknotted embedding
$e:T^p\to R^{p+2}$, where $T^p$ is the $p$-dimensional torus. Those two lower bounds are known to be sharp.
This a joint work of Ding-Liu-Wang-Yao.