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Geometry of Groups and Moduli Spaces

群几何和模空间

基本信息

批准号：
16204005
负责人：
MORITA Shigeyuki
金额：
$ 10.48万
依托单位：
UNIVERSITY OF TOKYO
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

项目摘要

We investigated mapping class group of surfaces and moduli space of Riemann surfaces as well as related groups and moduli spaces mainly from the viewpoint of topology. In a joint work with Penner, the representative constructed a canonical 1-cocycle on the Teichmuller space. Kawazumi, together with Penner and Bene, realized higher Johnson homomorphisms combinatorially on the Teichmuller space. Furuta developed theory of Floer homotopy type associated to the Seiberg-Witten theory and Tsuboi obtained a remarkable result concerning the structure of real analytic diffeomorphism group of manifolds. Fujiwara, Kohno, and Matsumoto obtained deep results in combinatorial structure of hyperbolic as well as the mapping class group, study of configuration spaces and arithmetic mapping class group, respectively. Mitsumatsu, Kitano, Akita with-Kawazumi, Hirose, and Murakami obtained interesting results, respectively, in the studies of 3-dimensional contact geometry, twisted Alexander polynomials, integral Morita-Mumford classes, mapping class group of surfaces in 4-manifolds, and volumes in hyperbolic geometry. Also the representative studied the structure of the Lie algebra consisting of symplectic derivations of the tensor algebra, without unit, generated by the homology of surfaces and, as an application, constructed unstable homology classes of genus 1 moduli spaces. Based on these results, further study of our theme has come into view.

主要从拓扑学的角度研究了曲面的映射类群和黎曼曲面的模空间以及相关的群和模空间。在与Penner的合作中，代表在Teichmuller空间上构造了一个典型的1-上循环。Kawazumi，连同Penner和贝内，实现了更高的约翰逊同态组合的Teichmuller空间。古田发展理论的Floer同伦型相关联的Seiberg-Witten理论和坪井获得了显着的结果结构的真实的分析同胚群的流形。Fujiwara、Kohno和松本分别在双曲型的组合结构、映射类群、构形空间和算术映射类群的研究中获得了较深入的结果。Mitsumatsu，Kitano，秋田with-Kawazumi，Hirose和Murakami分别在三维接触几何，扭曲亚历山大多项式，积分Morita-Mumford类，4-流形中曲面的映射类群和双曲几何中的体积的研究中获得了有趣的结果。此外，该代表研究了由曲面的同调生成的张量代数的辛导子组成的李代数的结构，作为应用，构造了亏格为1的模空间的不稳定同调类。在此基础上，对本课题的进一步研究也就有了展望。