4-dimensional topology and topological field theory.

4 维拓扑和拓扑场论。

• 批准号: 11440021
• 负责人: FURUTA Mikio
• 金额: $ 5.18万
• 依托单位: University of Tokyo (2000)Kyoto University (1999)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B).
• 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/en/grant/KAKENHI-PROJECT-11440021/
• 关键词: 4-dimensional topology; hyperkahler quotient; Seiberg-Witten theory; gauge theory; topological field theory; index theorem; monopole equation; moduli space; モノボール方程式

1 The main purpose of the present research is to develope K-theoretical aspect of Seiberg-Witten theory.(1) Kametani and Furuta extended the notions of Pin structures and e-invariants to obtain an application to the stable-homotopy Seiberg-Witten invariants. They actually showed an inequality stronger than the 11/8-inequality under some condition on the cup product on H^1.(2) Minami showed "G-join theorem" and apply it to the stable-homotopy Seiberg-Witten invariants and improved the 10/8-theorem for spin 4-manifolds.(3) Ue, Furuta and Y.Fukumoto investigated properties of of the w-invariant, which they defined, and obtained some applications to Seiberg-fibered homology 3-spheres.(4) Konnno calculated the cohomology ring of the moduli spaces of polygons as examples of hyper-kahler quotients.(5) Kotani obtained a central limit theorem for magnetic transition operators on a crystal lattice.2 The main members of the present research regularly attended a series of conferences to discuss the following topics.(1) Mochizuki explained his work on r-spin structure. Furuta pointed out a possibility of a construction in 2-dimensional topology which is parallel to the stable homotopy version of the Seiberg-Witten invariants.(2) M.Morishita (Univ. Kanazawa) was invited to explain his idea about a similarity between number theory and 3-dimensional topology. We discussed Mazur's unpublished work on a similarity between general knots and fibered knots when the fundamental groups of the knot complements are completed in some way.

1 本研究的主要目的是发展 Seiberg-Witten 理论的 K 理论方面。(1) Kametani 和 Furuta 扩展了 Pin 结构和 e-不变量的概念，以获得稳定同伦 Seiberg-Witten 不变量的应用。他们实际上在 H^1 上的杯积上在某些条件下表现出了比 11/8 不等式更强的不等式。(2) Minami 展示了“G-连接定理”并将其应用于稳定同伦 Seiberg-Witten 不变量并改进自旋4流形的10/8定理。(3)Ue、Furuta和Y.Fukumoto研究了他们定义的w-不变量的性质，并得到了一些应用到 Seiberg 纤维同调 3-球体。(4) Konnno 计算了多边形模空间的上同调环作为超卡勒商的例子。(5) Kotani 获得了晶格上磁跃迁算符的中心极限定理。2目前研究的主要成员定期参加一系列会议，讨论以下主题：(1)望月新一解释了他在r-自旋结构方面的工作。 Furuta 指出了二维拓扑结构的可能性，它与 Seiberg-Witten 不变量的稳定同伦版本平行。(2) M.Morishita (金泽大学) 受邀解释他关于数之间相似性的想法理论和3维拓扑。我们讨论了 Mazur 未发表的关于当结补组的基本组以某种方式完成时一般结和纤维结之间的相似性的工作。

项目成果

M.Furuta,Y.Kametani,H.Matsue and N.Minami: "Stable-homotopy Seiberg-Witten invariants and Pin bordisms"Preprint.

M.Furuta、Y.Kametani、H.Matsue 和 N.Minami：“稳定同伦 Seiberg-Witten 不变量和 Pin bordism”预印本。

M.Kotani: "A central limit theorem for magnetic transition operators on a crystal lattice"Preprint.

M.Kotani：“晶格上磁跃迁算子的中心极限定理”预印本。

N.Norihiko: "The G-join theorem -an unbased G-Freaudenthal theorem"(preprint).

N.Norihiko：“G-join 定理 - 一个无基础的 G-Freaudenthal 定理”（预印本）。

M.Kotani: "The pressure and higher correlations for an Anosov diffeomorphism, with T.Sunada"Ergod.Th.Dynam.Sys.. (to appear).

M.Kotani：“阿诺索夫微分同胚的压力和更高的相关性，与 T.Sunada”Ergod.Th.Dynam.Sys..（即将出现）。

M.Kotani: "A note on asymptotic expansions for closed geodesics in homology class"preprint.

M.Kotani：“关于同调类中封闭测地线渐近展开的注释”预印本。