Topological study of Riemann surfaces through infinite-dimensional Lie algebras

2015 Fiscal Year Final Research Report

• Project/Area Number: 24340010
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: The University of Tokyo
• Principal Investigator: Kawazumi Nariya 東京大学, 数理(科)学研究科(研究院), 准教授 (30214646)
• Co-Investigator(Renkei-kenkyūsha): TADOKORO Yuuki 木更津工業高等専門学校, 准教授 (10435414) ; SATOH Takao 東京理科大学, 理学部第二部, 准教授 (70533256) ; SATO Masatoshi 東京電機大学, 未来科学部, 准教授 (10632010) ; KUNO Yusuke 津田塾大学, 学芸学部, 准教授 (80632760)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Keywords: リーマン面 / ゴールドマン・リー代数 / 写像類群 / 正則ホモトピー / トゥラエフ余括弧積 / 榎本佐藤トレース / 発散コサイクル / 無限次元リー代数

Outline of Final Research Achievements

We introduced a new regular homotopy version of the Goldman-Turaev Lie bialgebra, where null-homotopic loops are reserved. This lead us a new close relationship among the Goldman-Turaev Lie bialgebra, the Enomoto-Satoh trace and the Kashiwara-Vergne problem. (Kawazumi)
We gave a tensorial description of the completed Goldman Lie algebra of a compact connected surface with non-empty boundary. As colloraries, we obrained ``the infinitesimal Dehn-Nielsen theorem” and a geometric formulation of the Johnson homomorphisms in the case where the boundary is not connected.(Kawazumi-Kuno)
We showed that the homological Goldman Lie algebra is finitely generated, and that the smallest number of its generating system is 2g+2.(Kawazumi-Kuno-Toda)

Free Research Field

位相幾何学