Tuesday Seminar on Topology (October, 2010 -- January, 2011)

[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 24, 2011
Information :　
Toshitake Kohno
Nariya Kawazumi

October 12 -- Room 056, 16:30 -- 18:00

Andrei Pajitnov (Univ. de Nantes, The University of Tokyo)

Asymptotics of Morse numbers of finite coverings of manifolds

Abstract: Let X be a closed manifold; denote by m(X) the Morse number of X (that is, the minimal number of critical points of a Morse function on X). Let Y be a finite covering of X of degree d.
In this survey talk we will address the following question posed by M. Gromov: What are the asymptotic properties of m(N) as d goes to infinity?
It turns out that for high-dimensional manifolds with free abelian fundamental group the asymptotics of the number m(N)/d is directly related to the Novikov homology of N. We prove this theorem and discuss related results.

October 19 -- Room 002, 16:30 -- 18:00

Jinseok Cho (Waseda University)

Optimistic limits of colored Jones invariants

Abstract: Yokota made a wonderful theory on the optimistic limit of Kashaev invariant of a hyperbolic knot that the limit determines the hyperbolic volume and the Chern-Simons invariant of the knot. Especially, his theory enables us to calculate the volume of a knot combinatorially from its diagram for many cases.
We will briefly discuss Yokota theory, and then move to the optimistic limit of colored Jones invariant. We will explain a parallel version of Yokota theory based on the optimistic limit of colored Jones invariant. Especially, we will show the optimistic limit of colored Jones invariant coincides with that of Kashaev invariant modulo 2\pi^2. This implies the optimistic limit of colored Jones invariant also determines the volume and Chern-Simons invariant of the knot, and probably more information. This is a joint-work with Jun Murakami of Waseda University.

October 26 -- Room 056, 17:00 -- 18:00

Kazuo Habiro (RIMS, Kyoto University)

Quantum fundamental groups and quantum representation varieties for 3-manifolds

Abstract: We define a refinement of the fundamental groups of 3-manifolds and a generalization of representation variety of the fundamental group of 3-manifolds. We consider the category $H$ whose morphisms are nonnegative integers, where $n$ corresponds to a genus $n$ handlebody equipped with an embedding of a disc into the boundary, and whose morphisms are the isotopy classes of embeddings of handlebodies compatible with the embeddings of the disc into the boundaries. For each 3-manifold $M$ with an embedding of a disc into the boundary, we can construct a contravariant functor from $H$ to the category of sets, where the object $n$ of $H$ is mapped to the set of isotopy classes of embedding of the genus $n$ handlebody into $M$, compatible with the embeddings of the disc into the boundaries. This functor can be regarded as a refinement of the fundamental group of $M$, and we call it the quantum fundamental group of $M$. Using this invariant, we can construct for each co-ribbon Hopf algebra $A$ an invariant of 3-manifolds which may be regarded as (the space of regular functions on) the representation variety of $M$ with respect to $A$.

November 2 -- Room 056, 16:30 -- 18:00

Daniel Ruberman (Brandeis University)

Periodic-end manifolds and SW theory.

Abstract: We study an extension of Seiberg-Witten invariants to 4-manifolds with the homology of S^1 \times S^3. This extension has many potential applications in low-dimensional topology, including the study of the homology cobordism group. Because b_2^+ =0, the usual strategy for defining invariants does not work--one cannot disregard reducible solutions. In fact, the count of solutions can jump in a family of metrics or perturbations. To remedy this, we define an index-theoretic counter-term that jumps by the same amount. The counterterm is the index of the Dirac operator on a manifold with a periodic end modeled at infinity by the infinite cyclic cover of the manifold. This is joint work with Tomasz Mrowka and Nikolai Saveliev.

November 9 -- Room 056, 17:00 -- 18:00

Ken'ichi Ohshika (Osaka University)

Characterising bumping points on deformation spaces of Kleinian groups

Abstract: It is known that components of the interior of a deformation space of a Kleinian group can bump, and a component of the interior can bump itself, on the boundary of the deformation space. Anderson-Canary-McCullogh gave a necessary and sufficient condition for two components to bump. In this talk, I shall give a criterion for points on the boundary to be bumping points.

November 16 -- Room 056, 16:30 -- 18:00

Noboru Ito (Waseda University)

On a colored Khovanov bicomplex

Abstract: We discuss the existence of a bicomplex which is a Khovanov-type complex associated with categorification of a colored Jones polynomial. This is an answer to the question proposed by A. Beliakova and S. Wehrli. Then the second term of the spectral sequence of the bicomplex corresponds to the Khovanov-type homology group. In this talk, we explain how to define the bicomplex. If time permits, we also define a colored Rasmussen invariant by using another spectral sequence of the colored Jones polynomial.

November 30 -- Room 056, 16:30 -- 18:00

Nobuhiro Nakamura (The University of Tokyo)

Pin^-(2)-monopole equations and intersection forms with local coefficients of 4-manifolds

Abstract: We introduce a variant of the Seiberg-Witten equations, Pin^-(2)-monopole equations, and explain its applications to intersection forms with local coefficients of 4-manifolds. The first application is an analogue of Froyshov's results on 4-manifolds which have definite forms with local coefficients. The second one is a local coefficient version of Furuta's 10/8-inequality. As a corollary, we construct nonsmoothable spin 4-manifolds satisfying Rohlin's theorem and the 10/8-inequality.

December 7 -- Room 056, 16:30 -- 18:00

Raphael Ponge (The University of Tokyo)

Diffeomorphism-invariant geometries and noncommutative geometry.

Abstract: In many geometric situations we may encounter the action of a group $G$ on a manifold $M$, e.g., in the context of foliations. If the action is free and proper, then the quotient $M/G$ is a smooth manifold. However, in general the quotient $M/G$ need not even be Hausdorff. Furthermore, it is well-known that a manifold has structure invariant under the full group of diffeomorphisms except the differentiable structure itself. Under these conditions how can one do diffeomorphism-invariant geometry?
Noncommutative geometry provides us with the solution of trading the ill-behaved space $M/G$ for a non-commutative algebra which essentially plays the role of the algebra of smooth functions on that space. The local index formula of Atiyah-Singer ultimately holds in the setting of noncommutative geometry. Using this framework Connes and Moscovici then obtained in the 90s a striking reformulation of the local index formula in diffeomorphism-invariant geometry.
An important part the talk will be devoted to reviewing noncommutative geometry and Connes-Moscovici's index formula. We will then hint to on- going projects about reformulating the local index formula in two new geometric settings: biholomorphism-invariant geometry of strictly pseudo-convex domains and contactomorphism-invariant geometry.

December 14 -- Room 056, 16:30 -- 18:00

Kenneth Schackleton (IPMU)

On the coarse geometry of Teichmueller space

Abstract: We discuss the synthetic geometry of the pants graph in comparison with the Weil-Petersson metric, whose geometry the pants graph coarsely models following work of Brock’s. We also restrict our attention to the 5-holed sphere, studying the Gromov bordification of the pants graph and the dynamics of pseudo-Anosov mapping classes.

January 11 -- Room 056, 16:30 -- 18:00

Nariya Kawazumi (The University of Tokyo)

The Chas-Sullivan conjecture for a surface of infinite genus

Abstract: Let \Sigma_{\infty,1} be the inductive limit of compact oriented surfaces with one boundary component. We prove the center of the Goldman Lie algebra of the surface \Sigma_{\infty,1} is spanned by the constant loop. A similar statement for a closed oriented surface was conjectured by Chas and Sullivan, and proved by Etingof. Our result is deduced from a computation of the center of the Lie algebra of oriented chord diagrams. If time permits, the Lie bracket on the space of linear chord diagrams will be discussed. This talk is based on a joint work with Yusuke Kuno (Hiroshima U./JSPS).

January 25 -- Room 056, 16:30 -- 17:30

Chikara Haruta (The University of Tokyo)

On unknotting of surface-knots with small sheet numbers

Abstract: A connected surface smoothly embedded in ${\mathbb R}^4$ is called a surface-knot. In particular, if a surface-knot $F$ is homeomorphic to the $2$-sphere or the torus, then it is called an $S^2$-knot or a $T^2$-knot, respectively. The sheet number of a surface-knot is an invariant analogous to the crossing number of a $1$-knot. M. Saito and S. Satoh proved some results concerning the sheet number of an $S^2$-knot. In particular, it is known that an $S^2$-knot is trivial if and only if its sheet number is $1$, and there is no $S^2$-knot whose sheet number is $2$. In this talk, we show that there is no $S^2$-knot whose sheet number is $3$, and a $T^2$-knot is trivial if and only if its sheet number is $1$.