Tuesday Seminar on Topology (October, 2006 -- January, 2007)

[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 21, 2007
Information :　
Toshitake Kohno
Nariya Kawazumi

October 10 -- Room 056, 16:30 -- 18:00

Elmar Vogt (Freie Universitat Berlin)

Estimating Lusternik-Schnirelmann Category for Foliations: A Survey of Available Techniques

Abstract: The Lusternik-Schnirelmann category of a space $X$ is the smallest number $r$ such that $X$ can be covered by $r + 1$ open sets which are contractible in $X$. For foliated manifolds there are several notions generalizing this concept, all of them due to Helen Colman. We are mostly concerned with the concept of tangential Lusternik-Schnirelmann category (tangential LS-category). Here one requires a covering by open sets $U$ with the following property. There is a leafwise homotopy starting with the inclusion of $U$ and ending in a map that throws for each leaf $F$ of the foliation each component of $U \cap F$ onto a single point. A leafwise homotopy is a homotopy moving points only inside leaves. Rather than presenting the still very few results obtained about the LS category of foliations, we survey techniques, mostly quite elementary, to estimate the tangential LS-category from below and above.

October 17 -- Room 056, 16:30 -- 18:00

Arnaud Deruelle (University of Tokyo）

Networking Seifert Fibered Surgeries on Knots (joint work with Katura Miyazaki and Kimihiko Motegi)

Abstract: We define a Seifert Surgery Network which consists of integral Dehn surgeries on knots yielding Seifert fiber spaces; here we allow Seifert fiber space with a fiber of index zero as degenerate cases. Then we establish some fundamental properties of the network. Using the notion of the network, we will clarify relationships among known Seifert surgeries. In particular, we demonstrate that many Seifert surgeries are closely connected to those on torus knots in Seifert Surgery Network. Our study suggests that the network enables us to make a global picture of Seifert surgeries.　

October 24 -- Room 056, 16:30 -- 18:00

Marco Zunino　(University of Tokyo，JSPS）

A review of crossed G-structures

Abstract: We present the definition of "crossed structures" as introduced by Turaev and others a few years ago. One of the original motivations in the introduction of these structures and of the related notion of a "Homotopy Quantum Field Theory" (HQFT) was the extension of Reshetikhin-Turaev invariants to the case of flat principal bundles on 3-manifolds. We resume both this aspect of the theory and other applications in both algebra and topology and we present our results on the algebraic structures involved.

November 10 (Friday) -- Room 118, 17:40 -- 19:00

Kazuhiro Hikami　(University of Tokyo）

WRT invariant for Seifert manifolds and modular forms

Abstract: We study the SU(2) Witten-Reshetikhin-Turaev invariant for Seifert manifold. We disuss a relationship with the Eichler integral of half-integral modular form and Ramanujan mock theta functions.

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

Masamichi Takase　(Shinshu University）

High-codimensional knots spun about manifolds

Abstract: The spinning describes several methods of constructing higher-dimensional knots from lower-dimensional knots. The original spinning (Emil Artin, 1925) has been generalized in various ways. Using one of the most generalized forms of spinning, called "deform-spinning about a submanifold" (Dennis Roseman, 1989), we analyze in a geometric way Haefliger's smoothly knotted (4k-1)-spheres in the 6k-sphere.

November 28 -- Room 056, 17:00 -- 18:00

Kazuo Akutagawa (Tokyo University of Science)

The Yamabe constants of infinite coverings and a positive mass theorem

Abstract:\ The {\it Yamabe constant} $Y(M, C)$ of a given closed conformal manifold $(M, C)$ is defined by the infimum of the normalized total-scalar-curavarure functional $E$ among all metrics in $C$. The study of the second variation of this functional $E$ led O.Kobayashi and Schoen to independently introduce a natural differential-topological invariant $Y(M)$, which is obtained by taking the supremum of $Y(M, C)$ over the space of all conformal classes. This invariant $Y(M)$ is called the {\it Yamabe invariant} of $M$. For the study of the Yamabe invariant, the relationship between $Y(M, C)$ and those of its conformal coverings is important, the case when $Y(M, C)> 0$ particularly. When $Y(M, C) \leq 0$, by the uniqueness of unit-volume constant scalar curvature metrics in $C$, the desired relation is clear. When $Y(M, C) > 0$, such a uniqueness does not hold. However, Aubin proved that $Y(M, C)$ is strictly less than the Yamabe constant of any of its non-trivial {\it finite} conformal coverings, called {\it Aubin's Lemma}. In this talk, we generalize this lemma to the one for the Yamabe constant of any $(M_{\infty}, C_{\infty})$ of its {\it infinite} conformal coverings, under a certain topological condition on the relation between $\pi_1(M)$ and $\pi_1(M_{\infty})$. For the proof of this, we aslo establish a version of positive mass theorem for a specific class of asymptotically flat manifolds with singularities.

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

Maxim Kazarian (Steklov Math. Institute)

Thom polynomials for maps of curves with isolated singularities (joint with S. Lando)

Abstract: Thom (residual) polynomials in characteristic classes are used in the analysis of geometry of functional spaces. They serve as a tool in description of classes Poincar\'e dual to subvarieties of functions of prescribed types. We give explicit universal expressions for residual polynomials in spaces of functions on complex curves having isolated singularities and multisingularities, in terms of few characteristic classes. These expressions lead to a partial explicit description of a stratification of Hurwitz spaces.

December 19 -- Room 056, 16:30 -- 18:30

Keiichi Sakai　(University of Tokyo）

Poisson structures on the homology of the spaces of knots

Abstract: We study the homological properties of the space $K$ of (framed) long knots in $\R^n$, $n>3$, in particular its Poisson algebra structures. We had known two kinds of Poisson structures, both of which are based on the action of little disks operad. One definition is via the action on the space $K$. Another comes from the action of chains of little disks on the Hochschild complex of an operad, which appears as $E^1$-term of certain spectral sequence converging to $H_* (K)$. The main result is that these two Poisson structures are the same. We compute the first non-trivial example of the Poisson bracket. We show that this gives a first example of the homology class of $K$ which does not directly correspond to any chord diagrams.


Kyohei Yoshida (University of Tokyo）

On projections of pseudo-ribbon sphere-links.

Abstract: Suppose $F$ is an embedded closed surface in $R^4$. We call $F$ a pseudo-ribbon surface link if its projection is an immersion of $F$ into $R^3$ whose self-intersection set $\Gamma(F)$ consists of disjointly embedded circles. H. Aiso classified pseudo-ribbon sphere-knots ($F$ is a sphere.) when $\Gamma(F)$ consists of less than 6 circles. We classify pseudo-ribbon sphere-links when $F$ is two spheres and $\Gamma(F)$ consists of less than 7 circles.

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

Hirofumi Sasahira　(University of Tokyo）

An SO(3)-version of 2-torsion instanton invariants

Abstract: We construct invariants for simply connected, non-spin $4$-manifolds using torsion cohomology classes of moduli spaces of ASD connections on $SO(3)$-bundles. The invariants are $SO(3)$-version of Fintushel-Stern's $2$-torsion instanton invariants. We show that this $SO(3)$-torsion invariant of $2CP^2 \# -CP^2$ is non-trivial, while it is known that any invariants of $2CP^2 \# - CP^2$ coming from the Seiberg-Witten theory are trivial since $2CP^2 \# -CP^2$ has a positive scalar curvature metric.


Yamaguchi Yoshikazu (University of Tokyo）

On the non-acyclic Reidemeister torsion for knots

Abstract: The Reidemeister torsion is an invariant of a CW-complex and a representation of its fundamental group. We consider the Reidemeister torsion for a knot exterior in a homology three sphere and a representation given by the composition of an SL(2, C)- (or SU(2)-) representation of the knot group and the adjoint action to the Lie algebra. We will see that this invariant is expressed by the differential coefficient of the twisted Alexander invariant of the knot and investigate some properties of the invariant by using this relation.

January 23 -- Room 056, 16:30 -- 18:30

Fuminori Nakata 　(University of Tokyo）

The twistor correspondence for self-dual Zollfrei metrics ----their singularities and reduction

Abstract: C. LeBrun and L. J. Mason investigated a twistor-type correspondence between indefinite conformal structures of signature (2,2) with some properties and totally real embeddings from RP^3 to CP^3. In this talk, two themes following LeBrun and Mason are explained.
First we consider an additional structure: the conformal structure is equipped with a null surface foliation which has some singularity. We establish a global twistor correspondence for such structures, and we show that a low dimensional correspondence between some quotient spaces is induced from this twistor correspondence.
Next we formulate a certain singularity for the conformal structures. We generalize the formulation of LeBrun and Mason's theorem admitting the singularity, and we show explicit examples.


Ryo Ohashi (University of Tokyo）

On the homology group of Out(F_n)

Abstract: Suppose $F_n$ is the free group of rank $n$, $Out(F_n) = Aut(F_n)/Inn(F_n)$ the outer automorphism group of $F_n$. We compute $H_*(Out(F_n);\mathbb{Q})$ for $n\leq 6$ and conclude that non-trivial classes in this range are generated by Morita classes $\mu_i \in H_{4i}(Out(F_{2i+2});\mathbb{Q})$. Also we compute odd primary part of $H^*(Out(F_4);\mathbb{Z})$.

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

John F. Duncan (Harvard University)

Elliptic genera and some finite groups

Abstract: Recent developments in the representation theory of sporadic groups suggest new formulations of `moonshine' in which Jacobi forms take on the role played by modular forms in the monstrous case. On the other hand, Jacobi forms arise naturally in the study of elliptic genera. We review the use of vertex algebra as a tool for constructing the elliptic genus of a suitable vector bundle, and illustrate connections between this and vertex algebraic representations of certain sporadic simple groups.