Complex analytic approach towards topology studies on the mapping class ganups for surfaces

曲面映射类组合拓扑研究的复杂分析方法

• 批准号: 14540065
• 负责人: KAWAZUMI Nariya
• 金额: $ 2.69万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540065/
• 关键词: Riemann surface; moduli space; mapping class group; Magnus expansion; Morita-Mumford class; Stasheff associahedron; automorphism group of a free group; Johnson homomorphism; マグナス表現; スタシェフのアソシアヘドロン; 超楕円的写像類群; 調和積分論; 擬等角変形; 積分周期; 森田マシフォード類

We discovered a close relation between Stasheff associahedrons and (generalized) Magnus expansions of a free group. A certain part of the twisted Morita-Mumford classes can be extended to the automorphism group of a free group. It is parametrized by Stasheff associahedrons "infinitesimally" and "combinatorially" how the extended Johnson maps are far from true group homomorphisms.We extended our theory on harmonic Magnus expansions to the universal family of Riemann surfaces. This yields another series of canonical 1 forms on the universal family than what we have already obtained on the moduli space. As a corollary, we obtained a proof that the first Jonson map and the (0,3)-twisted Morita-Mumford class coincides with each other as differential forms on the moduli space.The Magus representation of the automorphism group of a free group was constructed in an intrinsic manner. Here 'intrinsic' means 'with no use of Fox' free differentials.'We co-organized a workshop entitled "Toward the future of the topological study of manifolds" in November 2004.

我们发现斯塔谢夫联体面与自由群的（广义）马格努斯展开之间存在密切关系。扭曲的 Morita-Mumford 类的某一部分可以扩展到自由群的自同构群。它由斯塔舍夫关联面体“无穷小地”和“组合地”参数化扩展约翰逊映射如何远离真正的群同态。我们将调和马格努斯展开的理论扩展到黎曼曲面的普遍族。这产生了普适族上的另一系列规范 1 形式，而不是我们在模空间上已经获得的形式。作为推论，我们得到了第一个 Jonson 映射和 (0,3)-扭曲 Morita-Mumford 类在模空间上作为微分形式彼此一致的证明。自由群自同构群的 Magus 表示为以内在的方式构建。这里的“本征”意味着“不使用 Fox 自由微分”。 2004 年 11 月，我们共同组织了一个题为“走向流形拓扑研究的未来”的研讨会。

项目成果

Vertex-face correspondence of Boltzmann weights related to sl(m|n)

与 sl(m|n) 相关的玻尔兹曼权重的点面对应

• 发表时间: 2004
• 作者: Youichi Shibukawa

Word representaion of cords on a punctured plane

刺穿平面上绳索的文字表示

• 发表时间: 2005
• 作者: Yukio Matsumoro; Seiichi Kamada
• 通讯作者: Seiichi Kamada

Weierstrass points and Morita-Mumford classes on hyperelliptic mapping class group

超椭圆映射类群上的 Weierstrass 点和 Morita-Mumford 类

• 发表时间: 2002
• 作者: Nariya Kawazumi

Vertex-face correspondence of Boltzmann weights related to sl(m|n)

与 sl(m|n) 相关的玻尔兹曼权重的点面对应

• 发表时间: 2004
• 作者: Youichi Shibukawa

Generators for the tautological algebra of the moduli space of curves

曲线模空间同义反复代数生成器

• 发表时间: 2003
• 作者: Shigeyuki Morita