Mathematical Sciences: Topics in Low-Dimensional Topology

数学科学：低维拓扑专题

• 批准号: 9204331
• 负责人: Lee Mosher
• 金额: $ 14.22万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-08-15 至 1997-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9204331&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topics; Low; Dimensional

The study of 3-manifolds is often enriched by imposing extra structure on the manifold: geometric structure such as a Riemannian metric of negative curvature, or topological-dynamical structure such as a flow or lamination. The two investigators will continue research into 3-manifold supporting laminations and pseudo-Anosov flows. They will also undertake new research into the structure of 3-manifolds and more general spaces which are not negatively curved. In a joint project, Mosher and Oertel will explore a new technique for analyzing such spaces. Any such space can be made to support a certain kind of 2-dimensional measured lamination. The properties of this lamination will be studied and used in an attempt to shed light on the general structure of these spaces. In a separate project, Mosher will continue a study of Thurston's homology norm for a 3-manifold, and techniques for computing this norm using pseudo-Anosov flows. He will also pursue a computer study of ends of hyperbolic 3- manifolds. In still another separate project, Oertel will continue research into a class of 3-manifolds called laminated manifolds. The first goal is to extend methods and theorems from the well-known class of Haken manifolds to the larger class of laminated manifolds. The second goal is to show that, in some sense, "most" 3-manifolds are laminated. 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 investigators are 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.

3-流形的研究通常通过在流形上施加额外的结构来丰富：几何结构，如负曲率的黎曼度量，或拓扑-动力结构，如流或分层。这两位研究人员将继续研究支持叠层和伪阿诺索夫流的3-流形。他们还将对三维流形和更一般的非负曲线空间的结构进行新的研究。在一个联合项目中，Mosher和Oertel将探索一种分析此类空间的新技术。任何这样的空间都可以被制造成支撑某种类型的二维测量叠层。将研究和使用这种叠层的特性，试图阐明这些空间的一般结构。在另一个单独的项目中，Mosher将继续研究3-流形上的瑟斯顿同调范数，以及使用伪Anosov流计算该范数的技术。他还将对双曲三维流形的末端进行计算机研究。在另一个单独的项目中，Oertel将继续研究一类称为叠层流形的3-流形。第一个目标是将方法和定理从著名的Haken流形推广到更大的层叠流形。第二个目标是证明，在某种意义上，“大多数”3-流形是层合的。一个令人惊讶的事实是，尽管我们生活在一个三维空间，也就是所谓的三维流形中，因此我们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，其中一些问题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距离概念的流形的情况有明显的相关性。