Surfaces in finite covers of 3-manifolds and aspects of the mapping class groups

3-流形的有限覆盖中的表面和映射类组的方面

• 批准号: 0707136
• 负责人: Nathan Dunfield
• 金额: $ 27.99万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-01 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0707136&HistoricalAwards=false
• 关键词: Surfaces; finite; covers; manifolds; aspects

The PI will study the topology of 3-dimensional manifolds focusing on questions related to the Virtual Haken Conjecture. This conjecture posits that every closed 3-manifold with infinite fundamental group contains a surface whose fundamental group injects into that of the three manifold, if one passes to a finite cover. The ultimate goal of the proposal is to prove this conjecture. The PI will approach the question with two main themes: the use of random 3-manifolds to determine the most profitable avenues of attack, and the use of number theory to construct illuminating examples.A secondary focus of the work will be the mapping class groups of surfaces, in particular a study of the relationship between random walk harmonic measures and the Lebesgue measure on the space of projective measured laminations.The PI intends to work on one of the central questions in topology of three-dimensional manifolds using methods coming from several disciplines of mathematics, including topology, probability and number theory. He also intends to develod software to explore aspects of his research program. This software will be made available to other researchers via the web. Graduate students will be involved in this research.

PI将研究三维流形的拓扑结构，重点关注与虚拟哈肯猜想相关的问题。这个猜想假定每个具有无限基本群的闭3-流形包含一个曲面，如果一个曲面传递到有限覆盖，则该曲面的基本群注入到该3-流形的基本群中。该提案的最终目标是证明这一猜想。PI将从两个主题着手处理该问题：使用随机三维流形来确定最有利可图的攻击途径，并使用数论来构建启发性的例子。工作的第二个重点是曲面的映射类组，特别是研究了射影可测层空间上的随机游动调和测度与Lebesgue测度之间的关系，打算使用来自数学的几个学科，包括拓扑学，概率和数论的方法来研究三维流形拓扑学中的一个中心问题。他还打算开发软件来探索他的研究计划的各个方面。该软件将通过网络提供给其他研究人员。研究生将参与这项研究。