Tuesday Seminar on Topology (April -- July, 2010)

[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 July 13, 2010
Information :　
Toshitake Kohno
Nariya Kawazumi

April 13 -- Room 056, 16:30 -- 18:00

Christian Kassel (CNRS, Univ. de Strasbourg)

Torsors in non-commutative geometry

Abstract: G-torsors or principal homogeneous spaces are familiar objects in geometry. I'll present an extension of such objects to "non-commutative geometry". When the structural group G is finite, non-commutative G-torsors are governed by a group that has both an arithmetic component and a geometric one. The arithmetic part is given by a classical Galois cohomology group; the geometric input is encoded in a (not necessarily abelian) group that takes into account all normal abelian subgroups of G of central type. Various examples will be exhibited.

April 20 -- Room 056, 16:30 -- 18:00

Hélène Eynard-Bontemps (The University of Tokyo, JSPS)

Homotopy of foliations in dimension 3.

Abstract: We are interested in the connectedness of the space of codimension one foliations on a closed 3-manifold. In 1969, J. Wood proved the fundamental result:
Theorem: Every 2-plane field on a closed 3-manifold is homotopic to a foliation.
W. R. gave a new proof of (and generalized) this result in 1973 using local constructions. It is then natural to wonder if two foliations with homotopic tangent plane fields can be linked by a continuous path of foliations. A. Larcanché gave a positive answer in the particular case of "sufficiently close" taut foliations. We use the key construction of her proof (among other tools) to show that this is actually always true, provided one is not too picky about the regularity of the foliations of the path:
Theorem: Two C^\infty foliations with homotopic tangent plane fields can be linked by a path of C^1 foliations.

April 27 -- Room 056, 16:30 -- 18:00

Yoshiyuki Yokota (Tokyo Metropolitan University)

On the complex volume of hyperbolic knots

Abstract: In this talk, we give a formula of the volume and the Chern-Simons invariant of hyperbolic knot complements, which is closely related to the volume conjecture of hyperbolic knots. We also discuss the volumes and the Chern-Simons invariants of closed 3-manifolds obtained by Dehn surgeries on hyperbolic knots.

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

Nariya Kawazumi (The University of Tokyo)

The logarithms of Dehn twists

Abstract: We establish an explicit formula for the action of (non-separating and separating) Dehn twists on the complete group ring of the fundamental group of a surface. It generalizes the classical transvection formula on the first homology. The proof is involved with a homological interpretation of the Goldman Lie algebra. This talk is based on a jointwork with Yusuke Kuno (Hiroshima U./JSPS).

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

Naoyuki Monden (Osaka University)

On roots of Dehn twists

Abstract: Let $t_{c}$ be the Dehn twist about a nonseparating simple closed curve $c$ in a closed orientable surface. If a mapping class $f$ satisfies $t_{c}=f^{n}$ in mapping class group, we call $f$ a root of $t_{c}$ of degree $n$. In 2009, Margalit and Schleimer constructed roots of $t_{c}$. In this talk, I will explain the data set which determine a root of $t_{c}$ up to conjugacy. Moreover, I will explain the minimal and the maximal degree.

June 1 -- Room 056, 16:30 -- 18:00

Taro Asuke (The University of Tokyo)

On Fatou-Julia decompositions

Abstract: We will explain that Fatou-Julia decompositions can be introduced in a unified manner to several kinds of one-dimensional complex dynamical systems, which include the action of Kleinian groups, iteration of holomorphic mappings and complex codimension-one foliations. In this talk we will restrict ourselves mostly to the cases where the dynamical systems have a certain compactness, however, we will mention how to deal with dynamical systems without compactness.

June 15 -- Room 056, 16:30 -- 18:00

Kazuhiro Ichihara (Nihon University)

On exceptional surgeries on Montesinos knots (joint works with In Dae Jong and Shigeru Mizushima)

Abstract: I will report recent progresses of the study on exceptional surgeries on Montesinos knots. In particular, we will focus on how homological invariants (e.g. khovanov homology, knot Floer homology) on knots can be used in the study of Dehn surgery.

June 29 -- Room 056, 16:30 -- 18:00

Takahiro Kitayama (The University of Tokyo)

Non-commutative Reidemeister torsion and Morse-Novikov theory

Abstract: For a circle-valued Morse function of a closed oriented manifold, we show that Reidemeister torsion over a non-commutative formal Laurent polynomial ring equals the product of a certain non-commutative Lefschetz-type zeta function and the algebraic torsion of the Novikov complex over the ring. This gives a generalization of the results of Hutchings-Lee and Pazhitnov on abelian coefficients. As a consequence we obtain Morse theoretical and dynamical descriptions of the higher-order Alexander polynomials.

July 6 -- Room 056, 17:00 -- 18:00

Akira Kono (Kyoto University)

On the cohomology of free and twisted loop spaces

Abstract: A natural extension of cohomology suspension to a free loop space is constructed from the evaluation map and is shown to have a good properties in cohomology calculation. This map is generalized to a twisted loop space. As an application, the cohomology of free and twisted loop space of classifying spaces of compact Lie groups, including certain finite Chevalley groups is calculated.

July 13 -- Room 056, 16:30 -- 18:00

Marion Moore (University of California, Davis)

High Distance Knots in closed 3-manifolds

Abstract: Let M be a closed 3-manifold with a given Heegaard splitting. We show that after a single stabilization, some core of the stabilized splitting has arbitrarily high distance with respect to the splitting surface. This generalizes a result of Minsky, Moriah, and Schleimer for knots in S^3. We also show that in the complex of curves, handlebody sets are either coarsely distinct or identical. We define the coarse mapping class group of a Heeegaard splitting, and show that if (S,V,W) is a Heegaard splitting of genus greater than or equal to 2, then the coarse mapping class group of (S,V,W) is isomorphic to the mapping class group of (S, V, W). This is joint work with Matt Rathbun.

July 20 -- Room 056, 17:00 -- 18:00

Keiko Kawamuro (University of Iowa)

A polynomial invariant of pseudo-Anosov maps

Abstract: Thurston-Nielsen classified surface homomorphism into three classes. Among them, the pseudo-Anosov class is the most interesting since there is strong connection to the hyperbolic manifolds. As an invariant, the dilatation number has been known. In this talk, I will introduce a new invariant of pseudo-Anosov maps. It is an integer coefficient polynomial, which contains the dilatation as the largest real root and is often reducible. I will show properties of the polynomials, examples, and some application to knot theory. (This is a joint work with Joan Birman and Peter Brinkmann.)

July 27 -- Room 056, 16:30 -- 18:00

Ayumu Inoue (Tokyo Institute of Technology)

Quandle homology and complex volume (Joint work with Yuichi Kabaya)

Abstract: For a hyperbolic 3-manifold M, the complex value (Vol(M) + i CS(M)) is called the complex volume of M. Here, Vol(M) denotes the volume of M, and CS(M) the Chern-Simons invariant of M. In 2004, Neumann defined the extended Bloch group, and showed that there is an element of the extended Bloch group corresponding to a hyperbolic 3-manifold. He further showed that we can compute the complex volume of the manifold by evaluating the element of the extended Bloch group. To obtain an element of the extended Bloch group corresponding to a hyperbolic 3-manifold, we have to find an ideal triangulation of the manifold and to solve several equations. On the other hand, we show that the element of the extended Bloch group corresponding to the exterior of a hyperbolic link is obtained from a quandle shadow coloring. It means that we can compute the complex volume combinatorially from a link diagram.