EAPSI: An Investigation of Closed Surfaces in 3-manifolds via Character Varieties

EAPSI：通过特征变异研究 3 流形中的闭合曲面

• 批准号: 1713920
• 负责人: Charles Katerba
• 金额: $ 0.54万
• 依托单位: Katerba Charles W
• 依托单位国家: 美国
• 项目类别: Fellowship Award
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-06-01 至 2018-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1713920&HistoricalAwards=false
• 关键词: EAPSI; Investigation; Closed; Surfaces; manifolds

To understand the structure of a 3-dimensional shape, or 3-manifold, it often helps to study certain surfaces in the 3-manifold that encode important information. Culler-Shalen theory provides the most general tools for constructing such surfaces and has been instrumental in the resolution of many well-known questions. These special surfaces split naturally into two classes: those with boundary and those without. Much of Culler-Shalen theory's power comes from a deep understanding of the surfaces with boundary that one can construct with these tools. On the other hand, very little is known about the associated surfaces without boundary. In this project, we will use a novel algebraic perspective on Culler-Shalen theory to study the associated surfaces without boundary. Our work will resolve numerous problems concerning Culler-Shalen theory itself and will also open the door to applying the theory to an even broader class of questions about 3-manifolds. This research will be conducted in collaboration wth Dr. Stephan Tillman, a leading expert on Culler-Shalen theory and 3-manifolds, at the University of Sydney in Sydney, Australia.More specifically, Culler-Shalen theory uses the algebraic geometry of a 3-manifold's character variety and a construction, due to Stallings, to build essential surfaces in the manifold. The A-polynomial and the Culler-Shalen norm both determine precisely which boundary slopes arise in this fashion when the manifold's boundary consists of a single torus. However, not every boundary slope is detected by Culler-Shalen theory. We will address an analogous question concerning the detected closed essential surfaces using a module-theoretic perspective on character varieties developed by Chesebro. In particular, we hope to construct an in nite family of hyperbolic 3-manifolds with torus boundary containing closed essential surfaces that are not detected by the character variety. We will also explore the connection between the singular slopes of a closed surface and the bounded surfaces which are weakly detected by the character variety. This award, under the East Asia and Pacific Summer Institutes program, supports summer research by a U.S. graduate student and is jointly funded by NSF and the Australian Academy of Science.

为了理解三维形状或三维流形的结构，研究三维流形中编码重要信息的某些表面通常会有所帮助。Culler-Shalen理论为构造这样的曲面提供了最通用的工具，并在解决许多众所周知的问题方面发挥了重要作用。这些特殊的表面自然地分成两类：有边界的和没有边界的。Culler-Shalen理论的力量很大程度上来自于对可以用这些工具构建的具有边界的表面的深刻理解。另一方面，人们对无边界的相关曲面知之甚少。在这个项目中，我们将使用Culler-Shalen理论的一种新的代数视角来研究无边界的相关曲面。我们的工作将解决有关Culler-Shalen理论本身的许多问题，并将为将该理论应用于更广泛的关于3-流形的问题打开大门。这项研究将与澳大利亚悉尼大学的Culler-Shalen理论和3流形的主要专家Stephan Tillman博士合作进行。更具体地说，Culler-Shalen理论使用3-流形的特征变化的代数几何和由于Stallings的构造来构建流形中的基本表面。a -多项式和Culler-Shalen范数都精确地确定了流形边界由单个环面组成时哪种边界斜率以这种方式出现。然而，Culler-Shalen理论并不能检测到所有的边界斜率。我们将使用由Chesebro开发的字符变体的模理论观点来解决关于检测到的封闭本质曲面的类似问题。特别是，我们希望构造一个具有环面边界的双曲3-流形族，其中包含不被特征变化检测到的封闭本质曲面。我们还将探讨封闭曲面的奇异斜率与由特征变化弱检测的有界曲面之间的联系。该奖项由美国国家科学基金会和澳大利亚科学院共同资助，隶属于东亚和太平洋暑期研究所项目，支持一名美国研究生进行暑期研究。