スケジュール

スケジュール

7月30日

9:45 - 10:45 吉瀬流星

有限位相空間Khalimsky Circleの位相的複雑さ

11:00 - 12:00 武田雅広

The Steenrod problem and some graded Stanley Reisner rings

13:30 - 14:30 蔦谷充伸

Higher homotopy normalities in topological groups

14:45 - 15:45 若月駿

BV exactness and string brackets

16:00 - 17:00 栗林勝彦

Local systems in diffeology --Toward rational homotopy theory for diffeological spaces--

7月31日

9:45 - 10:45 Tong Yichen

Homotopy commutativity in Hermitian symmetric spaces

11:00 - 12:00 蓮井翔

The Stiefel-Whitney classes of moment-angle manifolds are trivial

アブストラクト

栗林勝彦

Local systems in diffeology --Toward rational homotopy theory for diffeological spaces--

By making use of Halperin's local systems over simplicial sets and the model structure of the category of diffeological spaces due to Kihara, we introduce a framework of rational homotopy theory for such smooth spaces with arbitrary fundamental groups. As a consequence, we have an equivalence between the homotopy categories of fibrewise rational diffeological spaces and an algebraic category of minimal local systems elaborated by Gómez-Tato, Halperin and Tanré.

Moreover, a spectral sequence converging to the singular de Rham cohomology of a diffeological adjunction space is constructed with a pullback of relevant local systems. In case of a stratifold obtained by attaching manifolds, the spectral sequence converges to the Souriau--de Rham cohomology algebra of the diffeological space. We also provide a commutative model for the Souriau--de Rham complex of the unreduced suspension of a connected closed manifold.

武田雅広

The Steenrod problem and some graded Stanley Reisner rings

The question “What kind of rings can be represented as singular cohomology rings of spaces?” is a classic problem in algebraic topology, posed by Steenrod. When rings are the polynomial rings, this problem was especially well studied by various approaches, and finally solved by Anderson and Grodal in 2008. In this talk, we will consider what kind of graded Stanley Reisner rings, as a generalization of polynomial rings, can be represented as cohomology rings by using a classical approach.

蔦谷充伸

Higher homotopy normalities in topological groups

Crossed module has important applications in homotopy theory, which is a generalization of normal subgroup. It is proved by Farjoun and Segev that the homotopy quotient of a topological crossed module is canonically a topological group. We introduce higher homotopy variants of crossed module called N_k(l)-maps (k,l ≥ 1), which are more restrictive classes of classical homotopy normlaities defined by McCarty and James. N_k(l)-map has a good characterization using fiberwise projective spaces. The homotopy quotient of an N_k(l)-map is shown to be an H-space if it has small LS category. As an application, we study when the inclusion SU(m) -> SU(n) is p-locally an N_k(l)-map.

Tong Yichen

Homotopy commutativity in Hermitian symmetric spaces

A fundamental problem on H-spaces is to find whether or not a given H-space is homotopy commutative. It is proved that the loop spaces of some homogeneous spaces are homotopy nilpotent, but we do not even know they are homotopy commutative or not. In this talk, we investigate the homotopy commtativity of loop spaces of irreducible Hermitian spaces case-by-case. The method also applies to compute the homotopy nilpotency of flag manifolds.

This is a joint work with Daisuke Kishimoto and Masahiro Takeda.

蓮井翔

The Stiefel-Whitney classes of moment-angle manifolds are trivial

If a moment-angle complex Z_K is a smooth manifold, we can easily see that Z_K is null-cobordant (i.e. Z_K=¥partial M for some manifold with boundary M) and therefore the Stiefel--Whitney numbers of Z_K are trivial. This observation naturally leads us to ask the question: Are the Stiefel--Whitney classes of a moment-angle manifold also trivial? In this talk, I'd like to show that this problem can be solved affirmatively. Note that, by introducing the notion of Stiefel--Whitney classes for topological manifolds due to Fadell, we can consider this problem for the moment-angle manifolds which are topological manifolds, not only for smooth ones. For such cases, the problem is also solved affirmatively. Moreover, we also consider the Stiefel-Whitney classes of a partial quotient of a moment-angle manifold.

吉瀬流星

有限位相空間Khalimsky Circleの位相的複雑さ

A finite space is a topological space consisting of finitely many points. Finite spaces are not discrete in general, and their homotopy theory has its own interest. In this talk, we determine the topological complexity of the Khalimsky circles, which are previously studied by Tanaka. As a consequence, we prove a conjecture of Tanaka on the topological complexity of a finite space affirmatively.

若月駿

BV exactness and string brackets

The string bracket is a Lie bracket on the S^1-equivariant homology of the free loop space of an oriented closed manifold, which is introduced by Chas and Sullivan. In this talk, we explain methods to compute them when the manifold is BV exact. This method can be applied to many examples, since any rationally formal space (or more generally a space admitting positive weights) is BV exact. We also give non-examples of BV exact spaces, which are found by a computer-assisted method. This is a joint work with Katsuhiko Kuribayashi, Takahito Naito, and Toshihiro Yamaguchi.