Tuesday Seminar on Topology (September, 2011 -- February, 2012)

[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 6, 2012
Information :　
Toshitake Kohno
Nariya Kawazumi

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

Clara Löh (Universität Regensburg)

Functorial semi-norms on singular homology

Abstract: Functorial semi-norms on singular homology add metric information to homology classes that is compatible with continuous maps. In particular, functorial semi-norms give rise to degree theorems for certain classes of manifolds; an invariant fitting into this context is Gromov's simplicial volume. On the other hand, knowledge about mapping degrees allows to construct functorial semi-norms with interesting properties; for example, so-called inflexible simply connected manifolds give rise to functorial semi-norms that are non-trivial on certain simply connected spaces.

October 4 -- Room 056, 16:30 -- 18:00

Yoshifumi Matsuda (The University of Tokyo)

Relatively quasiconvex subgroups of relatively hyperbolic groups

Abstract: Relative hyperbolicity of groups was introduced by Gromov as a generalization of word hyperbolicity. Motivating examples of relatively hyperbolic groups are fundamental groups of noncompact complete hyperbolic manifolds of finite volume. The class of relatively quasiconvex subgroups of a realtively hyperbolic group is defined as a genaralization of that of quasicovex subgroups of a word hyperbolic group. The notion of hyperbolically embedded subgroups of a relatively hyperbolic group was introduced by Osin and such groups are characterized as relatively quasiconvex subgroups with additional algebraic properties. In this talk I will present an introduction to relatively quasiconvex subgroups and discuss recent joint work with Shin-ichi Oguni and Saeko Yamagata on hyperbolically embedded subgroups.

October 11 -- Room 056, 17:00 -- 18:00

Gaël Meigniez (Univ. de Bretagne-Sud, Chuo Univ.)

Making foliations of codimension one, thirty years after Thurston's works

Abstract: In 1976 Thurston proved that every closed manifold M whose Euler characteristic is null carries a smooth foliation F of codimension one. He actually established a h-principle allowing the regularization of Haefliger structures through homotopy. I shall give some accounts of a new, simpler proof of Thurston's result, not using Mather's homology equivalence; and also show that this proof allows to make F have dense leaves if dim M is at least 4. The emphasis will be put on the high dimensions.

October 25 -- Room 056, 16:30 -- 18:00

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

Circle-valued Morse theory for complex hyperplane arrangements

Abstract: Let A be a complex hyperplane arrangement in an n-dimensional complex vector space V. Denote by H the union of the hyperplanes and by M the complement to H in V. We develop the real-valued and circle-valued Morse theory on M. We prove that if A is essential then M has the homotopy type of a space obtained from a finite n-dimensional CW complex fibered over a circle, by attaching several cells of dimension n. We compute the Novikov homology of M and show that its structure is similar to the homology with generic local coefficients: it vanishes for all dimensions except n.
This is a joint work with Toshitake Kohno.

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

Kiyoshi Takeuchi (University of Tsukuba)

Motivic Milnor fibers and Jordan normal forms of monodromies

Abstract: We introduce a method to calculate the equivariant Hodge-Deligne numbers of toric hypersurfaces. Then we apply it to motivic Milnor fibers introduced by Denef-Loeser and study the Jordan normal forms of the local and global monodromies of polynomials maps in various situations. Especially we focus our attention on monodromies at infinity studied by many people. The results will be explicitly described by the "convexity" of the Newton polyhedra of polynomials. This is a joint work with Y. Matsui and A. Esterov.

November 8 -- Room 056, 16:30 -- 18:00

Shoji Yokura (Kagoshima University)

Fiberwise bordism groups and related topics

Abstract: We have recently introduced the notion of fiberwise bordism. In this talk, after a quick review of some of the classical (co)bordism theories, we will explain motivations of considering fiberwise bordism and some results and connections with other known works, such as M. Kreck's bordism groups of orientation preserving diffeomorphisms and Emerson-Meyer's bivariant K-theory etc. An essential motivation is our recent work towards constructing a bivariant-theoretic analogue (in the sense of Fulton-MacPherson) of Levine-Morel's or Levine-Pandharipande's algebraic cobordism.

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

François Laudenbach (Univ. de Nantes)

Singular codimension-one foliations and twisted open books in dimension 3. (joint work with G. Meigniez)

Abstract: The allowed singularities are those of functions. According to A. Haefliger (1958), such structures on manifolds, called Γ₁-structures, are objects of a cohomological theory with a classifying space BΓ₁. The problem of cancelling the singularities (or regularization problem) arise naturally. For a closed manifold, it was solved by W.Thurston in a famous paper (1976), with a proof relying on Mather's isomorphism (1971): Diff^∞(R) as a discrete group has the same homology as the based loop space Ω BΓ₁⁺. For further extension to contact geometry, it is necessary to solve the regularization problem without using Mather's isomorphism. That is what we have done in dimension 3. Our result is the following.
Every Γ₁-structure ξ on a 3-manifold M whose normal bundle embeds into the tangent bundle to M is Γ₁-homotopic to a regular foliation carried by a (possibily twisted) open book.
The proof is elementary and relies on the dynamics of a (twisted) pseudo-gradient of ξ .
All the objects will be defined in the talk, in particular the notion of twisted open book which is a central object in the reported paper.

November 22 -- Room 056, 16:30 -- 18:00

Toshitake Kohno (The University of Tokyo)

Quantum and homological representations of braid groups

Abstract: Homological representations of braid groups are defined as the action of homeomorphisms of a punctured disk on the homology of an abelian covering of its configuration space. These representations were extensively studied by Lawrence, Krammer and Bigelow. In this talk we show that specializations of the homological representations of braid groups are equivalent to the monodromy of the KZ equation with values in the space of null vectors in the tensor product of Verma modules when the parameters are generic. To prove this we use representations of the solutions of the KZ equation by hypergeometric integrals due to Schechtman, Varchenko and others. In the case of special parameters these representations are extended to quantum representations of mapping class groups. We describe the images of such representations and show that the images of any Johnson subgroups contain non-abelian free groups if the genus and the level are sufficiently large. The last part is a joint work with Louis Funar.

November 29 -- Room 056, 17:00 -- 18:00

Athanase Papadopoulos (IRMA, Univ. de Strasbourg)

Mapping class group actions

Abstract: I will describe and present some rigidity results on mapping class group actions on spaces of foliations on surfaces, equipped with various topologies.

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

Mircea Voineagu (IPMU, The University of Tokyo)

Remarks on filtrations of the singular homology of real varieties.

Abstract: We discuss various conjectures about filtrations on the singular homology of real and complex varieties. We prove that a conjecture relating niveau filtration on Borel-Moore homology of real varieties and the image of generalized cycle maps from reduced Lawson homology is false. In the end, we discuss a certain decomposition of Borel-Haeflinger cycle map. This is a joint work with J.Heller.

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

Yoshihiko Mitsumatsu (Chuo University)

Leafwise symplectic structures on Lawson's Foliation on the 5-sphere

Abstract: We are going to show that Lawson's foliation on the 5-sphere admits a smooth leafwise symplectic sturcture. Historically, Lawson's foliation is the first one among foliations of codimension one which are constructed on the 5-sphere. It is obtained by modifying the Milnor fibration associated with the Fermat type cubic polynominal in three variables. Alberto Verjovsky proposed a question whether if the Lawson's foliation or slighty modified ones admit a leafwise smooth symplectic structure and/or a leafwise complex structure. As Lawson's one has a Kodaira-Thurston nil 4-manifold as a compact leaf, the question can not be solved simultaneously both for the symplectic and the complex cases. The main part of the construction is to show that the Fermat type cubic surface admits an `end-periodic' symplectic structure, while the natural one as an affine surface is conic at the end. Even though for the other two families of the simple elliptic hypersurface singularities almost the same construction works, at present, it seems very limited where a Stein manifold admits an end-periodic symplectic structure. If the time allows, we also discuss the existence of such structures on globally convex symplectic manifolds.

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

Takao Satoh (Tokyo University of Science)

On the Johnson cokernels of the mapping class group of a surface (joint work with Naoya Enomoto)

Abstract: In general, the Johnson homomorphisms of the mapping class group of a surface are used to investigate graded quotients of the Johnson filtration of the mapping class group. These graded quotients are considered as a sequence of approximations of the Torelli group. Now, there is a broad range of remarkable results for the Johnson homomorphisms. In this talk, we concentrate our focus on the cokernels of the Johnson homomorphisms of the mapping class group. By a work of Shigeyuki Morita and Hiroaki Nakamura, it is known that an Sp-irreducible module [k] appears in the cokernel of the k-th Johnson homomorphism with multiplicity one if k=2m+1 for any positive integer m. In general, however, to determine Sp-structure of the cokernel is quite a difficult preblem. Our goal is to show that we have detected new irreducible components in the cokernels. More precisely, we will show that there appears an Sp-irreducible module [1^k] in the cokernel of the k-th Johnson homomorphism with multiplicity one if k=4m+1 for any positive integer m.

February 21 -- Room 056, 16:30 -- 18:00

Masato Mimura (The University of Tokyo)

Property (TT)/T and homomorphism superrigidity into mapping class groups

Abstract: Mapping class groups (MCG's), of compact oriented surfaces (possibly with punctures), have many mysterious features: they behave not only like higher rank lattices (namely, irreducible lattices in higher rank algebraic groups); but also like rank one lattices. The following theorem, the Farb--Kaimanovich--Masur superrigidity, states a rank one phenomenon for MCG's: "every group homomorphism from higher rank lattices (such as SL(3,Z) and cocompact lattices in SL(3,R)) into MCG's has finite image."
In this talk, we show a generalization of the superrigidity above, to the case where higher rank lattices are replaced with some (non-arithmetic) matrix groups over general rings. Our main example of such groups is called the "universal lattice", that is, the special linear group over commutative finitely generated polynomial rings over integers, (such as SL(3,Z[x])). To prove this, we introduce the notion of "property (TT)/T" for groups, which is a strengthening of Kazhdan's property (T).
We will explain these properties and relations to ordinary and bounded cohomology of groups (with twisted unitary coefficients); and outline the proof of our result.