3-Manifold Geometry and Topology

3-流形几何和拓扑

• 批准号: 1105738
• 负责人: Ian Agol
• 金额: $ 25.22万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105738&HistoricalAwards=false
• 关键词: Manifold; Geometry; Topology

Under this grant, the PI will work on problems related to surfaces in 3-manifolds. The virtual Haken conjecture asks whether any aspherical closed 3-manifold has a finite-sheeted cover which contains a closed surface which injects on the fundamental group. The PI proposes to work on this conjecture making use of recent results of Kahn-Markovic and Wise. This would have ramifications for related conjectures, such as the virtual fibering conjecture, and would greatly elucidate the structure of 3-manifolds and their invariants, such as twisted Alexander polynomials. The PI also proposes to study the Thurston norm, and the relation between various surfaces realizing the Thurston norm in a minimal homology class via sutured manifold hierarchies. He hopes to use these relations to study various questions about the Thurston norm and its relation to other invariants of the 3-manifold.The mathematical study of 3-dimensional spaces goes back to the work of Poincare. Because classical physics describes our universe to be a 3-dimensional space, the classification of 3-dimensional spaces is an important mathematical endeavor, since it may have ramifications for the global structure of our universe. This project will pursue various aspects of the classification of 3-dimensional spaces. The principal and most ambitious project will study 2-dimensional objects inside of 3-dimensional spaces, and how one may resolve these in related "covering spaces". Studying these 2-dimensional objects has implications for the global structure of finite 3-dimensional spaces. The techniques of the project will have relations to other areas of mathematics such as group theory, geometry, and the study of 4-dimensional spaces.

根据这项补助金，PI将致力于与3-流形中的曲面相关的问题。虚Haken猜想提出了一个问题，即任何非球面闭三维流形是否有一个有限片覆盖，其中包含一个注入基本群的闭曲面。PI建议利用Kahn-Markovic和Wise的最新结果来研究这一猜想。这将对相关的理论产生影响，例如虚流形猜想，并将极大地阐明三维流形的结构及其不变量，例如扭曲的亚历山大多项式。PI还建议研究Thurston范数，以及通过缝合流形层次在最小同调类中实现Thurston范数的各种曲面之间的关系。他希望利用这些关系来研究各种问题的瑟斯顿规范及其关系到其他不变量的3-manifold.The数学研究三维空间可以追溯到工作的庞加莱。由于经典物理学将我们的宇宙描述为三维空间，因此三维空间的分类是一项重要的数学奋进，因为它可能对我们宇宙的整体结构产生影响。这个项目将追求三维空间分类的各个方面。 主要和最雄心勃勃的项目将研究三维空间内的二维物体，以及如何在相关的“覆盖空间”中解决这些问题。 研究这些二维物体对有限三维空间的整体结构有着重要意义。 该项目的技术将关系到其他数学领域，如群论，几何和四维空间的研究。