Abstracts

Special Lectures: Nobuaki Yagita (1st lecture) and Masaki Kameko (2nd lecture)

Cohomology of classifying spaces of compact Lie groups I, II

Starting with the review of the cohomology of the classifying space BG of a compact Lie group G whose integral cohomology has no torsion elements, we talk about the current status of the cases the integral homology of G has torsion elements. Furthermore, we talk about conjectures in the paper of Kono and Yagita (Trans. AMS, 1993) in conjunction with Totaro’s cycle map from the Chow ring of BG to the quotient of BP cohomology of BG. If time permits, we will talk about other open problems, e.g. Tezuka’s conjecture on the cohomology of finite Chevalley groups associated with G and that of the free loop space of BG.

The first speaker will explain mainly the cases of simply connected Lie groups. The second speaker will emphasize the role non-simply connected Lie groups should play in the cohomology of classifying spaces.

Sho Hasui

On the quasitoric manifolds over a simple polytope with one vertex cut

A quasitoric manifold is a smooth manifold with a good action of a compact torus for which the orbit space is a convex polytope. The most fundamental property of the quasitoric manifolds is that they are in one-to-one correspondence with the characteristic pairs, a kind of combinatorial objects, and we can get the classification up to (weakly) equivariant homeomorphism through this correspondence. However, when one try to advance to the classification up to homeomorphism, it turns out to be difficult since the combinatorial data seem to tell us almost nothing on the homeomorphisms which do not respect the torus actions. On the other hand, an interesting method was introduced in the joint work with Kuwata, Masuda, and Park on the classification of toric manifolds over a cube with one vertex cut, which gives a new way to construct homeomorphisms which are not (weakly) equivariant.

In this talk, we try to extend this method to more general situations from a homotopical viewpoint.

Katsuhiko Kuribayashi

The non-triviality of the whistle cobordism operation in string topology of classifying spaces

String topology gives fruitful structures to the loop homology of orientable closed manifolds, more general Gorenstein spaces. In particular, a result due to Chataur and Menichi asserts that the loop homology of the classifying space of a Lie group is endowed with the structure of a 2-dimensional topological quantum field theory (TQFT). Guldberg has proved that such a structure is generalized to that of a labeled open-closed TQFT. However, there are few calculations of labeled cobordism operations in the theory. In this talk, via an explicit calculation, we explain the non-triviality of a whistle cobordism operation with labels in the set of maximal closed subgroups of the given Lie group. It turns out that the open TQFT and closed one are not separated in general.

Ivan Limonchenko

On geometrical methods in the SU-bordism theory

In the first part of the talk we give a modernised exposition of the structure of the special unitary bordism ring based on a combination of the classical geometric methods of Conner-Floyd, Wall, and Stong with the Adams-Novikov spectral sequence and formal group law techniques that appeared after the fundamental 1967 paper of Novikov.

Toric topology provides us with a new geometric approach to calculations with SU-bordism, which allows to represent generators of the SU-bordism ring and other important SU-bordism classes by quasitoric manifolds and Calabi-Yau hypersurfaces in toric varieties. This approach will be discussed in the second part of this talk.

The talk is based on joint works with George Chernykh, Zhi Lu, and Taras Panov.

Takahiro Matsushita

Relative phantom maps and rational homotopy

This is a joint work with Daisuke Kishimoto.

A phantom map is a map from a CW-complex such that the restriction to every finite dimensional skeleton of X is null-homotopic. Recently, we and Iriye introduce a generalization of phantom maps, called relative phantom maps.

It is known that there is a close relationship between phantom maps and rational homotopy groups. For example, McGibbon and Roitberg show that if a map induces a surjection between their rational homotopy groups, then the map induces a surjection between the set of homotopy classes of phantom maps. We show some generalizations of this result to relative phantom maps.

Norihiko Minami

The "homework" of triangulated categories, and a hierachy of Higher Ruledness = Lower Rationality via generalized Bott towers

Algebraic topology and algebraic geometry have some common features from the viewpoint of triangulated categories. In our recent ArXiv preprint: http://arxiv.org/abs/1909.06538 (65 pages) to appear in Bousfield classes and Ohkawa's theorem, Springer Proceedings in Mathematics and Statistics, we presented a "Homework" of general triangulated categories.

In this talk, we start with this "Homework" and explain how its special case of SH(k), the Morel-Voevodsky A^1-stable homotopy category of schemes over k⊂C, especially the determination of its thick tensor ideals T(SH(k)) is important even to understand the ordinary stable homotopy category SH in algebraic topology. However, the determination of T(SH(k)) turns out to be a very difficult problem (though there is a recent effort by Ruth Joachmi, whose work will also appear in "Bousfield classes and Ohkawa's theorem", Springer Proceedings in Mathematics and Statistics,), and is regarded as one of the most fundamental open problems in the area. Still, we may take the Hopkins-Smith theorem on T(SH) as our lesson, and then we are immediately urged to look after "hierachies" in algebraic geometry.

At this stage, our talk shall change a gear toward our recent investigation of probably one of the most authentic "hierachies" in algebraic geometry, i.e. higher ruledness = lower rationality, which are birational invariant properties. In fact, we shall prove a sufficient criterion for some slightly stronger, NOT birational invariant, hierachy, which we call: uniregular-T^k-ruledness, where T^k is a k-dimensional generalized Bott tower, which nowadays many topologists would find familiar, and the dimension k reflects the hierarchy. When k=1, uniregular-T^1-ruledness is nothing but the ordinary uniruledness, but as k increases, the discrepancy between our uniregular-T^k-ruledness and the "classical" (though appears not have been written down explicitly in the past...) uni-k-ruledness appear to increase. Probably because of this, our sufficient criterion for uniregular-T^k-ruledness, when used as a sufficient criterion for uni-k-ruledness becomes less sharp as k increases, However, sill our sufficient criterion restricted to uni-k-ruledness for general k appears to be the first general criterion, and our sufficient criterion for uniregular-T^k-ruledness might be useful to produce some new examples for which the Hodge conjecture holds. Our proof depends upon the technique developed by Mori in his famous 79 Annals paper in which he solved the Hartshorne conjecture, and subsequent developments in algebraic geometry by Miyaoka, Kollar, Kebekus, Hwang-Mok, Druel, de Jong-Starr, Araujo-Castravet, and Taku Suzuki.

Keiichi Sakai

L-infinity algebras and graph cocycles

Rational cohomology groups of the long embedding spaces can be described as (a variant of) graph cohomology groups (G. Arone-V. Turchin).

In this talk I will explain how an L-infinity algebra with invariant scalar product gives rise to a graph cocycle that maps to a cocycle on the long knot space via configuration space integral.

Masahiro Takeda

The cohomology of the classifying spaces of certain gauge groups

A gauge group is the topological group of automorphisms of a principal bundle. For smooth bundles, its classifying space is homotopy equivalent to the moduli space of connections on the bundle. So the cohomology of the classifying space of a gauge group is important in its own right and also for applications. However, the integral cohomology has not been determined in a non-trivial case. I compute the integral cohomology ring of the classifying spaces of gauge groups of principal U(n)-bundles over the 2-sphere by generalizing the operation for free loop spaces, called the free double suspension.

Mitsunobu Tsutaya

Characterizations of homotopy fiber inclusion

It is well-known that a homotopy fiber sequence generated by a map G --> X extends twice to the right if and only if G admits a structure of a topological group and the map extends to an action of G on X up to homotopy equivalence. Similarly, a homotopy fiber sequence generated by a map H --> G extends three times to the right if and only if H and G admit structures of topological groups and the map is a homomorphism up to homotopy equivalence. But characterizations of "homotopy fiber inclusions" do not seem to be studied in detail. In this talk, we show some characterizations of homotopy fiber inclusions and some examples.

So Yamagata

Combinatorics of the Discriminantal arrangement

The Discriminantal arrangement was defined by Manin-Schechtman in 1989 as a generalization of the braid arrangement. Its combinatorics (i.e., intersection poset) is complicated and only partial description is known. In this talk, I will explain the connection between combinatorics of Discriminantal arrangement in codimension 2 and quadrics in Grassmannian.

Atsushi Yamaguchi

The Steenrod algebra from the group theoretical viewpoint

In the paper "The Steenrod algebra and its dual", J. Milnor determined the structure of the dual Steenrod algebra which is a graded commutative Hopf algebra of finite type. We consider the affine group scheme G represented by the dual Steenrod algebra. Then, G assigns a graded commutative algebra R* over a prime field of finite characteristic to a set of isomorphisms of the additive formal group law over R*, whose group structure is given by the composition of formal power series. The aim of this talk is to play with G and the notions of group theory by making use of this presentation of G(R*). For example, we give some families of subgroup schemes of G and estimate the length of the lower central series of a finite subgroup scheme of G. We also demonstrate some calculations on finite dimensional representations of small finite subgroup schemes of G and mention the restricted Lie algebra of G.

Masahiko Yoshinaga

Icosidodecahedron and Milnor fiber of arrangements

The relationship between combinatorial structure of a hyperplane arrangement and the topology of its Milnor fiber is largely unknown.

In the first part, I will survey Papadima-Suciu's framework for understanding the monodromy eigenspace of the Milnor fiber cohomology in terms of Aomoto complex with finite field coefficients. Then we show that the icosidodecahedral arrangement (an arrangement of 16 planes associated with the icosidodecahedron) provides a counterexample to a part of Papadima-Suciu's conjecture. The icosidodecahedral arrangement also provide the first example whose 1st homology of the Milnor fiber has a torsion. This talk is based on arXiv:1902.06256.