Tuesday Seminar on Topology (October, 2007 -- January, 2008)
[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 17, 2008
Information :@
Toshitake Kohno
Nariya Kawazumi
October 9 -- Room 056, 16:30 -- 18:00
Masayuki Asaoka (Kyoto University)
Classification of codimension-one locally free actions of the affine group of the real line.
Abstract:
By GA, we denote the group of affine and orientation-preserving transformations
of the real line. In this talk, I will report on classification of locally free action of
GA on closed three manifolds, which I obtained recently. In 1979, E.Ghys proved
that if such an action preserves a volume, then it is smoothly conjugate to a homogeneous
action. However, it was unknown whether non-homogeneous action exists. As a
consequence of the classification, we will see that the unit tangent bundle of a closed
surface of higher genus admits a finite-parameter family of non-homogeneous actions.
October 16 -- Room 056, 17:00 -- 18:00
Akito Futaki (Tokyo Institute of Technology)
Toric Sasaki-Einstein manifolds
Abstract:
A compact toric Sasaki manifold admits a Sasaki-Einstein metric if
and only if it is obtained by the Delzant construction from a toric
diagram of a constant height. As an application we see that the canonical
line bundle of a toric Fano manifold admits a complete Ricci-flat K\"ahler
metric.
October 23 -- Room 002, 16:30 -- 18:00
Jun O'Hara (Tokyo Metropolitan University)
Spaces of subspheres and their applications
Abstract:
The set of q-dimensional subspheres in S^n is a Grassmann manifold which has natural pseudo-Riemannian structure, and in some cases, symplectic structure as well. Both of them are conformally invariant.
I will explain some examples of their applications to geometric aspects of knots and links.
October 30 -- Room 056, 17:00 -- 18:00
Hiroshi Ohta (Nagoya University)
$L_{\infty}$ action on Lagrangian filtered $A_{\infty}$ algebras.
Abstract:
I will discuss $L_{\infty}$ actions on Lagrangian filtered
$A_{\infty}$ algebras@by cycles of the ambient symplectic
manifold. In the course of the construction,
I like to remark that the stable map compactification is not
sufficient in some case when we consider ones from genus zero
bordered Riemann surface. Also, if I have time, I like to discuss
some relation to (absolute) Gromov-Witten invariant and other
applications.
(This talk is based on my joint work with K.Fukaya, Y-G Oh and K. Ono.)
November 6 -- Room 056, 16:30 -- 18:00
Hiroki Kodama (The University of Tokyo)
Thurston's inequality and open book foliations
Abstract:
We will study codimension 1 foliations on 3-manifolds.
Thurston's inequality, which implies convexity of the foliation in
some sense, folds for Reebless foliations [Th]. We will discuss
whether the inequality holds or not for open book foliations.
[Th] W. Thurston: Norm on the homology of 3-manifolds, Memoirs of the
AMS, 339 (1986), 99--130.
November 20-- Room 056, 16:30 -- 18:00
Fumikazu Nagasato (Tokyo Institute of Technology)
A certain slice of the character variety of a knot group
and the knot contact homology
AbstractF
For a knot $K$ in 3-sphere, we can consider representations of
the knot group $G_K$ into $SL(2,\mathbb{C})$.
Their characters construct an algebraic set.
This is so-called the $SL(2,\mathbb{C})$-character variety of
$G_K$ and denoted by $X(G_K)$.
In this talk, we introduce a slice (a subset) $S_0(K)$ of $X(G_K)$.
In fact, this slice is closely related to the A-polynomial
and the abelian knot contact homology.
For example, the A-polynomial $A_K(m,l)$ of a knot $K$ is
a two-variable polynomial knot invariant defined by using
the character variety $X(G_K)$.
Then we can show that for any {\it small knot} $K$, the number of
irreducible components of $S_0(K)$ gives an upper bound of
the maximal degree of the A-polynomial $A_K(m,l)$ in terms of
the variable $l$.
Moreover, for any 2-bridge knot $K$, we can show that
the coordinate ring of $S_0(K)$ is exactly the degree 0
abelian knot contact homology $HC_0^{ab}(K)$.
We will mainly explain these facts.
November 27 -- Room 056, 16:30 -- 18:00
Atsushi Ishii (RIMS, Kyoto University)
A quandle cocycle invariant for handlebody-links
[joint work with Masahide Iwakiri (Osaka City University)]
Abstract:
A handlebody-link is a disjoint union of circles and a
finite trivalent graph embedded in a closed 3-manifold.
We consider it up to isotopies and IH-moves.
Then it represents an ambient isotopy class of
handlebodies embedded in the closed 3-manifold.
In this talk, I explain how a quandle cocycle invariant
is defined for handlebody-links.
December 4 -- Room 056, 16:30 -- 18:00
Hiroshi Konno (The University of Tokyo)
Morse theory for abelian hyperkahler quotients
Abstract:
In 1980's Kirwan computed Betti numbers of symplectic quotients by using Morse theory. In this talk, we develop this method to hyperkahler quotients by abelian Lie groups. In this method, many computations are much more simplified in the case of hyperkahler quotients than the case of symplectic quotients. As a result we compute not only the Betti numbers, but also the cohomology rings of abelian hyperkahler quotients.
December 11 -- Room 056
16:30 -- 17:30
Xavier Gómez-Mont (CIMAT, Mexico)
A Singular Version of The Poincaré-Hopf Theorem
Abstract:
The Poincaré-Hopf Theorem asserts that the Euler Characteristic of a compact manifold is the sum of the indices of any vector field on it with isolated singularities.
A hypersurface in real or complex number space may be considered as the limit of the smooth hypersurfaces obtained from nearby regular values. The singularity contains ghiddenh topology, which is unfolded by a smooth regeneration. At the singularity one has an algebraic invariant, the Jacobi Algebra, which is obtained by considering analytic functions modulo the partial derivatives. It contains topological information of the singularity.
One may consider vector fields tangent to a hypersurface with isolated singularities, and define topologically an index, which coincides with the sum of the Poincaré-Hopf indices of a regeneration of it tangent to a nearby smooth hypersurface.
I will explain how to compute the index of a vector field X tangent to an isolated hypersurface singularity V using Homological Algebra, as the Euler Characteristic of the homology of the complex obtained by contracting differential forms on V with the vector field X. The formula contains several terms, but the higher order terms may be translated from the invariants of the singular point to invariants in the Jacobi Algebra, making this translation a local version of the Poincaré-Hopf Theorem.
I will also explain how some of these ideas can be extended to complete intersections.
17:40 -- 18:40
Miguel A. Xicotencatl (CINVESTAV, Mexico)
Chen Ruan cohomology of cotangent orbifolds and Chas-Sullivan string topology
(Joint with: A. Gonzalez, E. Lupercio, C. Segovia, and B. Uribe)
Abstract:
At the end of 90's, two theories of topology were invented roughly at
the same time and attracted considerable interest in the mathematical
community. One is the Chas-Sullivan's loop product on the homology of
loop space and the second one is Chen-Ruan's stringy cohomology of
orbifold. It was an observation of Chen that inertia orbifold (which
carries Chen-Ruan cohomology) is the space of constant loops of an
orbifold. Therefore, two theories should interact. In this work we
show that for an interesting family of orbifolds, the virtual orbifold
cohomology, turns out to be a subalgebra of the homology of the loop
orbifold, and is isomorphic, as algebras, to the Chen-Ruan orbifold
cohomology of its cotangent orbifold.
December 18 -- Room 056, 16:30 -- 18:00
R.C. Penner (USC and Aarhus University)
Groupoid lifts of representations of mapping classes
Abstract: The "Ptolemy groupoid" is the fundamental path groupoid of the dual to the ideal cell decomposition of the decorated Teichmueller space of a punctured or bordered surface, and the "Torelli groupoid" is thesimilar discretization of the fundamental path groupoid of the quotient
by the Torelli subgroup of mapping classes that acts identically on the first integral homology of the surface. Mapping classes can be represented as appropriate elements of the Ptolemy groupoid and likewise for elements of the Torelli group in the Torelli groupoid.
A natural series of questions is to wonder which representations of mapping class groups, Torelli groups, and their subgroups can be lifted to the groupoid level. In a series of joint works with J. Andersen, A. Bene, N. Kawazumi, and S. Morita, we have given explicit lifts of a number of classical representations: The Johnson representations of the classical and higher Torelli groups
and the symplectic representation of the mapping class group all lift to the Torelli groupoid. Furthermore, the Nielsen representation of the mapping class group as automorphisms of a
free group lifts to the Ptolemy groupoid, and hence so too does any representation
of the mapping class group that factors through its action on the fundamental group of
the surface such as the Magnus representation. We shall survey these various groupoid lifts and discuss current and potential future applications.
January 15 -- Room 056, 16:30 -- 17:30
Shuichi Iida (The University of Tokyo)
Adiabatic limits of eta-invariants and the Meyer functions
Abstract: The Meyer function is the function defined on the hyperelliptic
mapping class group, which gives a signature formula for surface
bundles over surfaces.
In this talk, we introduce certain generalizations of the Meyer
function by using eta-invariants and we discuss the uniqueness of this
function and compute the values for Dehn twists.
January 29 -- Room 056, 16:30 -- 17:30
Yoshifumi Matsuda (The University of Tokyo)
The rotation number function on groups of circle diffeomorphisms
Abstract: Poincaré showed that the rotation number of an orientation-preserving
homeomorphism of the circle is rational if and only if it has a finite orbit.
In this talk, we explain that this fact can be generalized to certain groups of
orientation-preserving homeomorphisms of the circle.
In particular, we show that a nondiscrete group of orientation-preserving real
analytic diffeomorphisms of the circle has a finite image under the rotation
number function if and only if it has a finite orbit.
17:30 -- 18:30
Yasto Kimura (The University of Tokyo)
A Diagrammatic Construction
of Third Homology Classes of Knot Quandles
Abstract:There exists a family of third (quandle / rack) homology classes,
called the shadow (fundamental / diagram) classes,
of the knot quandle, which are obtained from
the shadow colourings of knot diagrams.
We will show the construction of these homology classes,
and also show their relation to the shadow quandle cocycle
invariants of knots and that to other third homology classes.