Abstracts

Takuma Imamura (Kyoto University)

Nonstandard homotopy theory

M. C. McCord (1972) developed a homology theory of topological spaces based on "hyperfinite" chains of "infinitesimal" simplices in the sense of Robinson's nonstandard analysis (NSA). This can be considered as the first application of NSA to algebraic topology. In this talk, we first explain the essence of NSA, and then introduce a homotopy theory of uniform spaces based on maps with "infinitesimal" discontinuity. After proving some basic properties of this homotopy, we prove a well-known (classical) theorem on the Vietoris-Cech homology theory of uniform spaces.

Masaki Kameko (Shibaura Institute of Technology)

Elementary abelian p-subgroups and the cohomology of the classifying space of a connected Lie group

For each odd prime p, we show that there exists a connected Lie group G such that the mod p cohomology of the classifying space BG is not detected by its elementary abelian p-subgroups.

Takahiro Matsushita (University of the Ryukyus)

A geometric interpretation of mixed commutator lengths

This is a joint work with Morimichi Kawasaki, Mitsuaki Kimura, and Masato Mimura.

Let G be a group and H a normal subgroup of G. A (G,H)-commutator is an element x in G such that there are g∈G and h∈H such that x = [g,h]. Let [G,H] be the subgroup of G generated by the set of (G,H)-commutators. Define the (G,H)-commutator length of x∈[G,H] to be the minimum number n such that there are n (G,H)-commutators whose product is x.

In the case of G = H, the (G,H)-commutator length is the commutator length which has been extensively studied. The following geometric interpretation of commutator length is well-known: Regard x∈[G,G] as a homotopy class of loops of BG. Since x∈[G,G] there is a compact oriented connected surface S with boundary S^1 and a map f: S -> BG such that the homotopy class of f|_{∂S} is x.

We give a similar geometric interpretation of the (G,H)-commutator lengths, and apply it to prove Bavard's duality theorem of G-invariant quasimorphisms on H.

Norihiko Minami (Nagoya Institute of Technology)

A sufficient criterion for some hierarchy stronger than: Higher uniruledness = Lower unirationality, applied to (weighted) complete intersections

Thanks to Morel-Voevodsky, classical homotopy theory had become a part of motivic homotopy theory over a filed k (⊂ C), whose extremely important problem is the determination of the thick ideals of the subtriangulated category consisting of compact objects. The case k is the number field would be the ultimate problem, but we may have to wait until the next century to see an affirmative solution to such a problem. In fact, even the case k=C is an unsolved difficult problem. In view of the Hopkins-Smith theorem in the classical stable homotopy category, the first step would find out and investigate a motivic homotopy theoretically useful hierarchy among algebraic varieties, but this is really difficult.

At the last homotopy theory symposium held at Sapporo, I reported a sufficient criterion, expressed in terms of somewhat complicated (numerical) inequalities, for an existence of a hierarchy described in terms of the generalized Bott towers and stronger than the one interpolating the classical algebro-geometrical concepts of uniruledness and unirationality among projective algebraic manifolds defined over C. This generalized Mori's famous uniruledness theorem of Fano manifolds to higher hierarchies.

While this result significantly generalizes previous approaches, and would be an ultimate result along this line, the apparent complexity might had obscured its usefulness. In this talk, I shall restrict this result to the case of weighed complete intersections, which were introduced again by Mori in his master thesis, and derived more explicit results. In the special case of complete intersections, our result becomes more transparent, and yields the number of times how many times VMRT in the sense of Hwang-Mok can be iterated (sometimes plus 1).

Yuya Miyata (Kyushu University)

On the topological complexity of a spherical space form

In 2010, Iwase and Sakai showed that the topological complexity of a space coincides with a fibrewise version of a L-S category for a fibrewise space over the space. In this presentation, we explain how to determine the topological complexity of S3/Q8 by using a method produced from the fibrewise viewpoint.

Syunji Moriya (Osaka Prefecture University)

On models for knot spaces

I talk about the space of knots in a manifold M of dimension 4 or more. In the former part of this talk we see the inclusion from the space of knots to the space of immersions from the circle to M induces isomorphism of pi_1 for some 4-manifolds M. This is done by observing a spectral sequence introduced in the preprint (arXiv: 2003.03815). In the latter part, we introduce a relatively simple diagram of Thom spaces and see the homotopy colimit of chain of this diagram is quasi-isomorphic to colimit of cochain Taylor tower of long knots in the Euclidean space of dimension 4 or more.

Takahito Naito (Nippon Institute of Technology)

The rational loop cutting coproduct and the Hodge decomposition

The loop cutting coproduct is a coproduct on the relative homology of free loop spaces of closed oriented manifolds. This operation was introduced by Sullivan in string topology and it is given by cutting of loops at the self intersection points. In this talk, we observe a behavior of the rational loop cutting coproduct in terms of the Hodge decomposition of the relative homology. Especially, we study the degree of the coproduct of pure manifolds in the decomposition by Sullivan models in rational homotopy theory.

Masahiro Takeda (Kyoto University)

Cohomology of the spaces of commuting elements

Let G be a compact connected Lie group, and let Hom(Z^m,G) denote the space of commuting m-tuples in G. Baird proved that the cohomology of Hom(Z^m,G) is identified with a certain ring of invariants of the Weyl group of G. Therefore the cohomology of Hom(Z^m,G) is important not only in topology but also in representation theory. We give a minimal generating set of the cohomology of Hom(Z^m,G) for G=SU(n), Sp(n), and the ring structure of the cohomology of Hom(Z^2,G) for G=SU(3), Sp(2), G_2. This talk is based on joint work with Daisuke Kishimoto.

Hiro Lee Tanaka (Texas State University)

Broken techniques for enriching Floer theory over spectra

The Fukaya category is an important invariant of symplectic manifolds. Since the work of Cohen-Jones-Segal, many attempts have been made to enrich the Fukaya category over spectra. (Even in the nicest cases, the Fukaya category is HZ-linear.) We will talk about ongoing work with Lurie to enrich Fukaya categories over spectra. Most of the talk will focus on the simplest case: Reconstructing Morse homology to recover the stable homotopy type of a compact manifold.

Takeshi Torii (Okayama University)

On duoidal infinity-categories

In stable homotopy theory, Hopf algebroids appear as cooperation rings of generalized cohomology theories. A duoidal category is a category equipped with two monoidal structures so that one is lax monoidal with respect to the other, and it is a setting in which we can define bialgebras. In this talk we will generalize duoidal categories in the setting of infinity-categories in order to consider Hopf algebroids in spectra, and give some examples of duoidal infinity-categories. In particular, we will show that a duoidal infinity-category can be obtained from a monoidal double infinity-category under some conditions. We will formulate bilax monoidal functors between duoidal infinity-categories, and define bialgebras in a duoidal infinity-category. If time permits, we will discuss generalizations of duoidal infinity-categories in various ways.

Atsushi Yamaguchi (Osaka Prefecture University)

On the fibered category of smooth maps and actions of diffeological groupoids

For a category C with finite limits, let M(C) be the category of morphisms of C, that is, M(C) is the category of functors from a finite category D to C, where D has two objects 0 and 1 and one non-identity morphism from 0 to 1. Then, it can be easily shown that the evaluation functor p from D to C at 1 is a fibered category whose inverse image functor obtained from a morphism f of C from X to Y given by the pull-backs along f of morphisms whose targets are Y. It is also easy to see that each inverse image functor of the fibered category p has a left adjoint, in other words p is a bifibered category.

On the other hand, for a groupoid object G=(G_0,G_1) in C and an object X of M(C) over G_0, the notion of a right action of G on X is defined which generalizes a right action of a group and we can consider the category Act(G) of right actions of G. If a morphism (f_0,f_1) of groupoids from H=(H_0,H_1) to G=(G_0,G_1) is given, the inverse image functor obtained from f_0 defines a functor from Act(G) to Act(H) by pulling back the action by f_1 and the left adjoint of the inverse image functor obtained from f_0 defines a left adjoint of the above functor from Act(G) to Act(H) under some mild conditions.

In this talk, we show that the inverse image functors of the fibered category p of morphisms of C also have right adjoints if C is the category of diffeological spaces. We also show that a morphism (f_0,f_1) of groupoids from H to G gives a right adjoint of the functor from Act(G) to Act(H) mentioned above. This fact applies to construct a diffeological fiber bundle with structure groupoid H from adiffeological fiber bundle with structure groupoid G.