Application of A_∞-methods on topological invariants

A_∞-方法在拓扑不变量上的应用

• 批准号: 15340025
• 负责人: IWASE Norio
• 金额: $ 3.84万
• 依托单位: Kyushu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-15340025/
• 关键词: A_∞-structure; L-S category; Lie group; Hopf invariant; co-Hopf space; A_∞構造; 特異点; L-Sカテゴリー; 球面束; 高次Hopf不変量; A_∞手法; L-Sカテゴリ数; Cone分解; 主束; 射影ユニタリー群

L-S category is defined by Lusternik and Schnirelmann to give a homotopy invariant of a topological space, which gives, for a manifold M, a lower bound of the number of the critical points of a C^∞-function on M. After the investigation of Ganea and many others, the following two problems were realised as open problems and listed in the book "Open Problems in Topology" by van Mill and M. Reed.[Problem 642] (Ganea conjecture) Is the L-S category of a space X × S^n equal to the L-S category of X plus 1?[Problem 643] Is the L-S category of a closed manifold greater than that of its once punctured submanifold?The higher Hopf invariant redefined on projective spaces of the loop space of a space, gave a criterion to determine L-S category, which implies negative answers to Problem 642 and Problem 643.In this research, extensively using the above idea, the present investigator completely determined the L-S category of a manifold which is the total space of a sphere-bundle over a sphere. As a result, we can construct manifolds as total spaces of S^2-bundles, which give counter examples to the Ganea conjecture. With Mamoru Mimura at Okayama University and Tetsu Nishimoto at Kinki Welfare University, he continued to investigate on the total space of a fibre bundle over a simply-connected suspension space or a non-simply-connected space.Combining this idea with the higher Hopf invariants, he together with Akira Kono at Kyoto University obtained a new upper bound for L-S category. They also defined a new lower bound for L-S category using the A_∞-method. These results yield the determination of L-S category of all compact simple Lie groups up to rank 4,except for Sp(4),F_4,PSp(3) and PSp(4). Using the A_∞-point of view, Kamata, Saeki, Sumi, Oda and Nishimoto have obtained various results.

L-S范畴由Lusternik和Schnirelmann定义，给出了拓扑空间的一个同伦不变量，它给出了流形M上C^∞函数在M上的临界点个数的下界。在对Ganea和其他许多人的研究之后，以下两个问题被实现为开放问题，并由van Mill和M. Reed在《拓扑中的开放问题》一书中列出。[问题642]（Ganea猜想）空间X × S^n的L-S范畴是否等于X + 1的L-S范畴？[问题643]一个封闭流形的L-S范畴是否大于它的一次穿刺子流形的L-S范畴？在空间的循环空间的射影空间上重新定义了更高的Hopf不变量，给出了确定L-S范畴的判据，给出了问题642和问题643的否定答案。在本研究中，广泛运用上述思想，完整地确定了流形的L-S范畴，即球束在球上的总空间。因此，我们可以将流形构造为S^2束的总空间，从而给出了Ganea猜想的反例。他与冈山大学的Mamoru Mimura和近木福利大学的Tetsu Nishimoto一起，继续研究单连通悬浮空间和非单连通空间上纤维束的总空间。他与京都大学的Akira Kono将这一思想与更高的Hopf不变量相结合，得到了L-S范畴的一个新的上界。他们还利用A_∞方法定义了L-S范畴的一个新的下界。这些结果得到了除Sp(4)、F_4、PSp(3)和PSp(4)外的所有4阶紧单李群的L-S类的确定。利用A_∞的观点，Kamata, Saeki, Sumi， Oda和Nishimoto得到了不同的结果。

项目成果

Gap modules for semidirect product groups

半直接产品组的间隙模块

• 发表时间: 2004
• 期刊: Kyushu Journal of Mathematics 58
• 作者: Toshio Sumi;Toshio Sumi
• 通讯作者: Toshio Sumi

L-S categories of simply-connected compact simple Lie groups of low rank

低阶单连紧单李群的 L-S 类

• 发表时间: 2004
• 期刊: Categorical decomposition techniques in algebraic topology (Isle of Skye, 2001), Progr. Math. 215
• 作者: L.A.Lucas;O.Saeki;Benaissa Bernoussi et al.;Osamu Saeki et al.;Walter Motta et al.;Benaissa Bernoussi et al.;Benaissa Bernoussi et al.;V.Blanl〓il et al.;Norio Iwase et al.
• 通讯作者: Norio Iwase et al.

On the characteristic numbers of the manifolds immersed in the multiple point set of a self-transverse immersion

关于自横向浸没多点集浸没流形的特征数

• 发表时间: 2006
• 期刊: Kyushu J. Math. 60
• 作者: Kamata;M.
• 通讯作者: M.

Lusternik-Schnirelmann category of a sphere-bundle over a sphere

球体上的球丛的 Lusternik-Schnirelmann 范畴

• 发表时间: 2003
• 期刊: Topology 42
• 作者: Iwase;N.
• 通讯作者: N.

Twisted tensor products related to the cohomology of the classifying spaces of loop groups. Mem. Amer. Math. Soc. No. 180

与环群分类空间的上同调相关的扭曲张量积。

• 发表时间: 2006
• 作者: Kuribayashi;K.;Mimura;M.;Nishimoto;T.
• 通讯作者: T.