Mathematical Sciences: Dynamical Systems in 3-dimensional Topology

数学科学：3 维拓扑中的动力系统

• 批准号: 9002587
• 负责人: Lee Mosher
• 金额: $ 5.92万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-07-01 至 1993-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9002587&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Dynamical; Systems; dimensional

The study of 3-dimensional topology is often enriched by imposing extra structure on 3-manifolds. A recent construction due to Gabai shows that a 3-manifold supports many interesting dynamical systems, namely pseudo-Anosov flows. The first two parts of this project aim towards showing that dynamical systems on a 3-manifold can be used to give complete information about an important topological invariant, the Thurston norm on homology. The main work of the first part will be to show that Gabai's flows are well-behaved enough to determine completely the Thurston norm. The second part will focus on explicitly constructing these flows, so as to have an efficient algorithm for computing the norm. A bonus will be a method for efficient computation of complete homeomorphism invariants of 3-manifolds which fiber over the circle. The third and final part of this project addresses a long range goal, to investigate the conjecture that every 3-manifold which is irreducible and contains a non-separating surface has a finite cover which fibers over the circle. Dynamical systems are potentially useful for this investigation, due to work of Fried. Mosher will attempt to prove the conjecture for some simple examples which are "as close as possible" to fibering without actually fibering. If successful, this may yield suggestions for an attack on the conjecture in general.

3 维拓扑的研究通常通过在 3 流形上施加额外的结构来丰富。 Gabai 最近的构造表明，3 流形支持许多有趣的动力系统，即伪阿诺索夫流。 该项目的前两部分旨在展示三流形上的动力系统可用于提供有关重要拓扑不变量（同源性瑟斯顿范数）的完整信息。 第一部分的主要工作是证明 Gabai 的流表现良好，足以完全确定瑟斯顿范数。 第二部分将重点关注显式构建这些流，以便有一个有效的算法来计算范数。 一个额外的好处是有效计算在圆上分布的 3 流形的完全同胚不变量的方法。 该项目的第三部分也是最后一部分解决了一个长期目标，研究以下猜想：每个不可约且包含非分离表面的 3 流形都具有在圆上纤维化的有限覆盖。 由于弗里德的工作，动力系统对于这项研究可能有用。 Mosher 将尝试用一些简单的例子来证明这个猜想，这些例子“尽可能接近”纤维化，但实际上并没有纤维化。 如果成功的话，这可能会产生对一般猜想进行攻击的建议。