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.

本质层理是最近定义的不可压缩曲面和余维--紧致三维流形中不含Reeb分量的一维叶理的推广。这两个更经典的对象在从同伦理论数据中获得关于3-流形的几何/拓扑信息方面都被证明是非常有用的，而本质分层已经开始显示出它自己是一个有价值的继承者。研究人员打算利用本质叠层来研究3-流形的基本群决定流形到同胚的程度，并研究流形是否具有双曲结构的问题。他还打算利用本质叠层的正规形的存在性来研究三维流形(主要是双曲三维流形)是否包含本质叠层的问题。令人惊讶的是，尽管我们生活在一个三维空间，也就是所谓的三维流形，因此对这样的几何对象有一种自然的直觉，但最终这并没有带给我们我们可能预期的那样远的距离，因为已经通过代数计算解决的问题在三维情况下仍然令人困惑。其中最著名的是世纪之交关于三维球体的著名的庞加莱猜想，其中确切地说，原始的三维情况是唯一仍然开放的。研究人员正在探索关于三维流形的各种问题，其中一些带有稍微奇怪的距离概念，即所谓的双曲线度量，但这些问题一次又一次地被证明与具有更熟悉的距离概念的流形的情况有明显的相关性。