Classifying spaces of Categories, Flore Homology and Homotopsy Theory of Infinite Complexes

范畴的分类空间、Flore 同调和无限复合体的同伦理论

• 批准号: 10440018
• 负责人: KONO Akira
• 金额: $ 2.05万
• 依托单位: KYOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10440018/
• 关键词: gauge group; free loop group; adjoint representation; Hopf space; Conley homology; Flore homology; Classifying space

The most important results of the research project are the following :(1) Topology of infinite dimensional Lie groups(2) Morse theory of infinite dimensional manifolds(3) Classifying space of categories and dynamical systemsFor (1) we determined homology ring of free loop groups for 1-connected Lie groups completely. Using the result we are nou determining the cohomology of the classifying spaces of them. For gauge groups, we showed the case when the base spaces are closed 1-connectedmanifolds and the structure group is SU(2), the homotopy type of the classifying spaces completely determine the boinotopy type of the base space and the isomorphism class of the bundles.For (2) K. Fukaya solved the Anord conjecture using the Gromov-Witten iavariants.For (3) we considered the Conley homology. We considerd a certain category and the cohomology of the classifying space of it is isomorphic to the Conley homology under some fine coditions.

(1)无限维李群的拓扑学(2)无限维流形的Morse理论(3)范畴空间和动力系统的分类对于(1)我们完全确定了1-连通李群的自由环群的同调环。利用这一结果，我们确定了它们的分类空间的上同调。对于规范群，我们证明了当基空间是闭的1-连通流形且结构群是SU(2)时，分类空间的同伦型完全决定了基空间的Boinotopy类型和丛的同构类.对于(2)K.Fukaya利用Gromov-Witten变元解决了Anord猜想.对于(3)我们考虑了Conley同调.我们认为某个范畴的分类空间的上同调在一定的条件下与Conley同调同构。

项目成果

K.Fukaya: "Arnold conjecture and Gromov-Witten invariant"Topology. 38. 933-1048 (1999)

K.Fukaya：《阿诺德猜想和格罗莫夫-维滕不变量》拓扑。

A. Moriwaki: "The continuity of Deligne pairing"internat. Math. Res. Notices. 1075-1066 (1999)

A. Moriwaki：“德利涅配对的连续性”国际。

H. Hamanaka: "Homology ring mod 2 of free loop groups of spinor groups"J. Pure and AppL. Algebra. 146. 267-282 (2000)

H. Hamanaka：“旋量群的自由环群的同源环模 2”J。

H. Nakajima: "University Lecture Ser. Vol. 18, Lectures on Hilbert schemes of points on surfaces"AMS. 132 (1999)

H. Nakajima：“大学讲座系列第 18 卷，关于曲面上点的希尔伯特方案的讲座”AMS。

A. Kono: "Adjoint action of a finite loop space II"Proc.Roy.Soc.Edinburgh Sect.A. 129. 773-785 (1999)

A. Kono：“有限循环空间 II 的伴随作用”Proc.Roy.Soc.Edinburgh Sect.A。