Low-Dimensional Topology via Bordered Floer Theory

通过有边弗洛尔理论的低维拓扑

• 批准号: 1711926
• 负责人: Jonathan Hanselman
• 金额: $ 10.26万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-15 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1711926&HistoricalAwards=false
• 关键词: Low; Dimensional; Topology; via; Bordered

This research project in low-dimensional topology investigates the global shape of various three-dimensional spaces and of knotted curves and surfaces within them. Since the space we inhabit is three-dimensional, this field has wide-ranging applications, from understanding the possible shapes of the universe to describing the knotting of polymers and DNA molecules. Understanding the properties of three-dimensional spaces requires an array of sophisticated tools from a range of disciplines within mathematics, including algebraic topology, geometry, analysis, and representation theory. Each tool for studying three-dimensional spaces carries with it a notion of which spaces are "simple" and which are "complicated." The primary goal of this project is to explore the relationship between different tools used to describe three-dimensional spaces and the corresponding notions of simplicity. Ultimately the investigator hopes to consolidate and deepen our knowledge of these spaces and to build bridges between a variety of mathematical fields.A key technical tool to be used in this investigation is bordered Heegaard Floer homology, a version of Heegaard Floer homology for 3-manifolds with boundary. In particular, the project centers on developing a new geometric interpretation of bordered Heegaard Floer invariants; this is a concrete realization of a deep connection between bordered Heegaard Floer homology and certain Fukaya categories. In the case of torus boundary, the invariants may be interpreted as decorated immersed curves in the boundary torus. This framework greatly simplifies computations and leads to proofs of interesting gluing results. This will be applied to the classification of L-spaces (3-manifolds that are "simple" with respect to Heegaard Floer homology). As one application, the investigator seeks to confirm a conjecture equating three measures of simplicity for 3-manifolds: being an L-space, having non-left-orderable fundamental group, and not admitting a co-orientable taut foliation. Other goals of the project include restricting the possible decompositions of L-space knots and relating Heegaard Floer homology to the complexity of the Jaco-Shalen-Johannson decomposition.

这个低维拓扑研究项目研究了各种三维空间以及其中的结曲线和曲面的全局形状。由于我们居住的空间是三维的，因此该领域具有广泛的应用，从理解宇宙可能的形状到描述聚合物和 DNA 分子的打结。理解三维空间的属性需要一系列数学学科中的复杂工具，包括代数拓扑、几何、分析和表示论。用于研究三维空间的每种工具都带有哪​​些空间是“简单”、哪些空间是“复杂”的概念。该项目的主要目标是探索用于描述三维空间的不同工具之间的关系以及相应的简单性概念。最终，研究者希望巩固和加深我们对这些空间的了解，并在各种数学领域之间建立桥梁。本次研究中使用的一个关键技术工具是有界 Heegaard Floer 同调，它是具有边界的 3 流形的 Heegaard Floer 同调的一个版本。特别是，该项目的重点是开发边界 Heegaard Floer 不变量的新几何解释；这是有界 Heegaard Floer 同源性与某些 Fukaya 范畴之间深层联系的具体实现。在环面边界的情况下，不变量可以被解释为边界环面中的装饰浸没曲线。该框架极大地简化了计算，并提供了有趣的粘合结果的证明。这将应用于 L 空间的分类（相对于 Heegaard Floer 同源性而言“简单”的 3 流形）。作为一个应用，研究者试图证实一个猜想，该猜想等同于 3-流形的三种简单性度量：是一个 L 空间，具有非左序基本群，并且不承认可同向拉紧叶状结构。该项目的其他目标包括限制 L 空间结的可能分解，并将 Heegaard Floer 同源性与 Jaco-Shalen-Johannson 分解的复杂性联系起来。