Tuesday Seminar on Topology (June -- July, 2020)
[Japanese] [Past Programs]
This is an online seminar on Zoom
Last updated September 10, 2020
Information :@
Nariya Kawazumi
Takahiro Kitayama
Takuya Sakasai
June 23 -- Online on Zoom.
Hokuto Konno (The University of Tokyo)
Gauge theory and the diffeomorphism and homeomorphism groups of 4-manifolds
Abstract: I will explain my recent collaboration with several groups that develops gauge theory for families
to extract difference between the diffeomorphism groups and the homeomorphism groups of 4-manifolds.
After Donaldsonfs celebrated diagonalization theorem,
gauge theory has given strong constraints on the topology of smooth 4-manifolds.
Combining such constraints with Freedmanfs theory, one may find many non-smoothable topological 4-manifolds.
Recently, a family version of this argument was started by T. Kato, N. Nakamura and myself,
and soon later it was developed also by D. Baraglia and his collaborating work with myself.
More precisely, considering gauge theory for smooth fiber bundles of 4-manifolds,
they obtained some constraints on the topology of smooth 4-manifold bundles.
Using such constraints, they detected non-smoothable topological fiber bundles of smooth 4-manifolds.
The existence of such bundles implies that there is homotopical difference between the diffeomorphism and homeomorphism groups of the 4-manifolds given as the fibers.
If time permits, I will also mention my collaboration with Baraglia which shows that a K3 surface gives a counterexample to the Nielsen realization problem in dimension 4.
This example reveals also that there is difference between the Nielsen realization problems asked in the smooth category and the topological category.
Slides
June 30 -- Online on Zoom.
Daniel Matei (IMAR Bucharest)
Homology of right-angled Artin kernels
Abstract: The right-angled Artin groups A(G) are the finitely presented groups associated to a finite simplicial graph G=(V,E),
which are generated by the vertices V satisfying commutator relations vw=wv for every edge vw in E.
An Artin kernel Nh(G) is defined by an epimorphism h of A(G) onto the integers.
In this talk, we discuss the module structure over the Laurent polynomial ring of the homology groups of Nh(G).
July 7 -- Online on Zoom.
Yuta Nozaki (Hiroshima University)
Abelian quotients of the Y-filtration on the homology cylinders via the LMO functor
Abstract: We construct a series of homomorphisms on the Y-filtration on the homology cylinders via the mod $_mathbb{Z}$ reduction of the LMO functor.
The restriction of our homomorphism to the lower central series of the Torelli group
does not factor through Morita's refinement of the Johnson homomorphism.
We use it to show that the abelianization of the Johnson kernel of a closed surface has torsion elements.
This is the joint work with Masatoshi Sato and Masaaki Suzuki.
Slides
July 14 [Joint with Lie Groups and Representation Theory Seminar] -- Online on Zoom.
Takayuki Okuda (Hiroshima University)
Kobayashi's properness criterion and totally geodesic submanifolds in locally symmetric spaces
Abstract: Let G be a Lie group and X a homogeneous G-space.
A discrete subgroup of G acting on X properly is called a discontinuous group for X.
We are interested in constructions and classifications of discontinuous groups for a given X.
It is well-known that if the isotropies of G on X are compact, any closed subgroup acts on X properly.
However, the cases where the isotropies are non-compact, the same claim does not hold in general.
Let us consider the case where G is a linear reductive.
In this situation, T. Kobayashi [Math. Ann. (1989)], [J. Lie Theory (1996)]
gave a criterion for the properness of the action on a homogeneous G-space X of closed subgroups in G.
In this talk, we consider homogeneous G-spaces of reductive types realized as families of totally geodesic submanifolds in non-compact Riemannian symmetric spaces.
As a main result, we give a translation of Kobayashi's criterion within the framework of Riemannian geometry.
In particular, for a torsion-free discrete subgroup of G,
the criterion can be stated in terms of totally geodesic submanifolds in the Riemannian locally symmetric space corresponding to the subgroup in G.
July 21 -- Online on Zoom.
17:00 -- 18:00
Sergei Burkin (The University of Tokyo)
Twisted arrow categories of operads and Segal conditions
Abstract: We generalize twisted arrow category construction from categories to operads,
and show that several important categories, including the simplex category Δ,
Segal's category Γ and Moerdijk--Weiss category Ω
are twisted arrow categories of operads.
Twisted arrow categories of operads are closely connected with Segal conditions,
and the corresponding theory can be generalized from multi-object associative algebras
(i.e. categories) to multi-object P-algebras for reasonably nice operads P.
18:00 -- 19:00
Dexie Lin (The University of Tokyo)
Monopole Floer homology for codimension-3 Riemannian foliation
Abstract: In this paper, we give a systematic study of Seiberg-Witten theory on closed oriented manifold with codimension-3 oriented Riemannian foliation.
Under a certain topological condition, we construct the basic monopole Floer homologies for a transverse spinc structure with a bundle-like metric, generic perturbation and a complete local system.
We will show that these homologies are independent of the bundle-like metric and generic perturbation.
The major difference between the basic monopole Floer homologies and the ones on manifolds is the necessity to use the complete local system to construct the monopole Floer homologies.
July 28 -- Online on Zoom.
Anderson Vera (RIMS, Kyoto University)
A double filtration for the mapping class group and the Goeritz group of the sphere
Abstract: I will talk about a double-indexed filtration of the mapping class group and of the Goeritz group of the sphere,
the latter is the group of isotopy classes of self-homeomorphisms of the 3-sphere which preserves the standard Heegaard splitting of S3.
In particular I will explain how this double filtration allows to write the Torelli group as a product of some subgroups of the mapping class group.
A similar study could be done for the group of automorphisms of a free group. (work in progress with K. Habiro)
Slides