Mathematical Sciences: Essential Laminations in 3-Manifolds

数学科学：3-流形中的基本叠片

• 批准号: 9203435
• 负责人: Mark Brittenham
• 金额: $ 3.44万
• 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: 1992
• 资助国家: 美国
• 起止时间: 1992-08-01 至 1995-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9203435&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Essential; Laminations; Manifolds

Essential laminations are a recently defined generalization of both the incompressible surface and the codimension-one foliation without Reeb components of a compact 3-manifold. Both of these more classical objects have proved remarkably useful in obtaining geometric/topological information about 3-manifolds from homotopy-theoretic data, and the essential lamination has already begun to show itself to be a worthy successor to both of its `parents.' The investigator intends to use essential laminations to study the extent to which the fundamental group of a 3-manifold determines the manifold up to homeomorphism, and also to study the question of whether or not the manifold admits a hyperbolic structure. He further intends to use the existence of normal forms for essential laminations to study the question of whether or not a 3-manifold (principally a hyperbolic 3-manifold) contains an essential lamination. It is a surprising fact that although we live in a three dimensional space, a so-called 3-manifold, and so are blessed with a natural intuition about such geometric objects, in the end this does not carry us as far as we might have expected, for questions which have been settled by algebraic calculations for higher dimensional manifolds still remain baffling in the 3-dimensional case. The most famous of these is the celebrated conjecture of Poincare from around the turn of the century concerning 3- dimensional spheres, where precisely the original 3-dimensional case is the only one still open. The investigator is pursuing a variety of questions about 3-dimensional manifolds, some with slightly strange notions of distance on them, so-called hyperbolic metrics, but time and time again these questions have been shown to have clear relevance to the case of manifolds with a more familiar notion of distance.

基本叠层是最近定义的不可压缩表面和余维一叶状结构的概括，没有紧凑 3 流形的 Reeb 组件。 事实证明，这两个更经典的对象对于从同伦理论数据获取有关 3-流形的几何/拓扑信息非常有用，并且基本叠层已经开始表明自己是其“父母”的有价值的继承者。 研究者打算利用本质叠层来研究3-流形的基本群在多大程度上决定流形直至同胚，并研究流形是否承认双曲结构的问题。 他进一步打算利用基本叠层的范式的存在性来研究3-流形（主要是双曲3-流形）是否包含基本叠层的问题。 令人惊讶的事实是，尽管我们生活在三维空间（即所谓的三流形）中，因此对此类几何对象拥有自然的直觉，但最终这并没有让我们走得尽可能远。正如预期的那样，对于通过高维流形的代数计算解决的问题在 3 维情况下仍然令人困惑。 其中最著名的是庞加莱在世纪之交关于 3 维球体的著名猜想，其中原始的 3 维情况恰恰是唯一仍然开放的情况。 研究人员正在研究有关 3 维流形的各种问题，其中一些带有稍微奇怪的距离概念，即所谓的双曲度量，但这些问题一次又一次地被证明与流形的情况有明显的相关性更熟悉的距离概念。