Hopf invariants and their application

Hopf不变量及其应用

• 批准号: 11640084
• 负责人: IWASE Norio
• 金额: $ 1.98万
• 依托单位: KYUSHU UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640084/
• 关键词: Hopf invariant; L-S Category; Square ring; loop space; A_∞-structure; squarering; A_<co>-構造; 有限ループ空間; P-torsion; Kudo-Araki operation; co-H-space; Ganea Conjecture

1. We obtained an affirmative answer to "the Ganea conjecture" on co-H-space of low dimensions, In the proof, we found an fundamental theorem on a co-Hopf space, However a counter-example was found in higher dimensions, The above fundamental theorem is used to show the example is a co-H-space.2. L-S category - a numrical invariant defined by Lusternik and Schnirelmann is studied in terms of "higher Hopf inuariuat" and "A_∞-structures on loop spaces". As a result, we found a counter exampled to Gamea's conjecture on L-S category as a simply connected closed manifold.3. when a space X is in "meta stable" range, End (ΣX) is not a ring but a "square" ring. We determine the structure of End (ΣX) completely in terms of a Hopf invariant, if X is a mapping cone of a sphere-mapping (in meta stable range).

1.我们在低维的co-H-空间上对Ganea猜想给出了肯定的回答，在证明中，我们发现了一个关于co-Hopf空间的基本定理，而在高维空间中发现了一个反例，利用上述基本定理说明了这个例子是一个co-H-空间.利用“高阶Hopf不变量”和“loop空间上的A_∞-结构”研究了Lusternik和Schnirelmann定义的数值不变量L-S范畴。结果，我们在L-S范畴上找到了一个反例，证明了Gamea的猜想是单连通闭流形.当空间X在“Meta稳定”范围内时，End（X）不是环而是“方”环.如果X是球映射的映射锥（在Meta稳定域），则完全用一个Hopf不变量确定End（End X）的结构.

项目成果

Norio Iwase: "The Ganea conjecture on co-H-spaces"Topology.

Norio Iwase：“共 H 空间上的 Ganea 猜想”拓扑。

H.Baaes,N.IWASE: "Square rings associated to elemmts in homocopy groups of spheres"Contemporary Mathematics.

H.Baaes，N.IWASE：“与球体同拷贝群中的元素相关的方环”当代数学。

N.Iwase: "The Ganea conjecture on co-H-spaces"Topology. (to appear).

N.Iwase：“共 H 空间上的 Ganea 猜想”拓扑。

N.Iwaseg: "The Ganea conjecture on co-H-space"Topology.

N.Iwaseg：“共 H 空间上的 Ganea 猜想”拓扑。

H.Baues,N.Iwase: "Square rings associated to elements in homotopy groups of spheres"Contemporary Mathematics.

H.Baues，N.Iwase：“与球体同伦群中的元素相关的方环”当代数学。