Mapping Class Group and Fiber Bundles

映射类组和光纤束

• 批准号: 2104346
• 负责人: Bena Tshishiku
• 金额: $ 32.21万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2104346&HistoricalAwards=false
• 关键词: Mapping; Class; Group; Fiber; Bundles

This project is concerned with the study of spaces (manifolds), especially in dimensions 2, 3, and 4. In the study of these objects, a fundamental role is played by the mapping class group, which captures the symmetries of a space and encodes gluing data for building new spaces, such as fiber bundles. The study of mapping class groups is a meeting ground for many areas including geometric group theory, dynamics, homotopy theory, algebraic geometry, and mathematical physics. This project focuses on using mapping class groups to understand geometric properties of fiber bundles. The project provides research training opportunities for graduate students and includes organizing the Brown-Yale Geometry and Topology Conference. The project has three main objectives. (1) Find new examples of non-flat bundles, especially surface bundles whose base is low-dimensional. Research in this direction will increase our understanding of the dynamics of surface homeomorphisms, the algebra of mapping class groups, and their relation. (2) Establish new connections between the dynamical properties of a pseudo-Anosov surface homeomorphism and the (non)arithmeticity of the associated hyperbolic mapping torus, thereby building on work of Thurston and Bowditch-Maclachlan-Reid. (3) Provide new examples of convex-cocompact subgroups of the mapping class group. This work is aimed at a conjecture of Farb-Mosher regarding purely pseudo-Anosov subgroups of mapping class groups, and it also has connections to 3-manifold topology. The project will also support the advising of students and the organization of a conference.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

这个项目是关于空间（流形）的研究，特别是在维度2,3和4。在对这些对象的研究中，映射类组扮演着一个基本的角色，它捕获空间的对称性，并为构建新空间（如光纤束）编码粘合数据。映射类群的研究是许多领域的交汇点，包括几何群论、动力学、同伦理论、代数几何和数学物理。这个项目的重点是使用映射类组来理解纤维束的几何特性。该项目为研究生提供了研究训练机会，并组织了布朗-耶鲁几何学和拓扑学会议。该项目有三个主要目标。(1)寻找新的非平坦束的例子，特别是低维基的曲面束。在这个方向上的研究将增加我们对表面同胚动力学、映射类群的代数及其关系的理解。(2)在Thurston和bowditch - maclachran - reid的工作基础上，建立了伪anosov曲面同纯的动力学性质与相关双曲映射环面的（非）算术性之间的新联系。(3)给出映射类群的凸紧子群的新例子。本文针对Farb-Mosher关于映射类群的纯伪anosov子群的一个猜想，它也与3流形拓扑有联系。该项目还将支持为学生提供建议和组织会议。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。