Mathematical Sciences: Representations of Three-Manifold Groups

数学科学：三流形群的表示

• 批准号: 9214499
• 负责人: Andrew Casson
• 金额: $ 18.09万
• 依托单位: University of California-Berkeley
• 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: 1992-12-01 至 1996-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9214499&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Representations; Three; Manifold

Andrew Casson will continue to study the geometrization problem for 3-manifolds. The recent solution of the Seifert Fiber Space Conjecture has reduced this to the case of irreducible 3- manifolds whose fundamental groups contain no free Abelian subgroups of rank two. Casson will concentrate on such manifolds which have infinite fundamental group; according to the "Geometrization Conjecture," these manifolds should all admit hyperbolic structures. He will seek simplified proofs of Thurston's now classical results establishing this conjecture for manifolds with non-trivial boundary, in the hope of developing methods which apply to the still unsolved case of manifolds without boundary. Robion Kirby will continue to search for topological interpretations of Witten's 3-manifold invariants for Sl(2,Z), and particularly for the case of finite coverings. 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 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.

安德鲁·卡森将继续研究 3-流形的问题 Seifert光纤的最新解决方案 空间猜想将其简化为不可约的3- 基本群不含自由阿贝尔的流形 二阶子群。 卡森将专注于这样的流形 有无限的基本群;根据 "几何化猜想，"这些流形都应该承认 双曲结构 他将寻求简化的证明， Thurston现在的经典结果建立了这个猜想， 流形与非平凡边界，希望发展 方法，适用于仍然未解决的情况下，流形没有 边界 Robion Kirby将继续寻找拓扑 对Sl（2，Z）的维滕三维流形不变量的解释，以及 特别是对于有限覆盖的情况。 这是一个令人惊讶的事实，虽然我们生活在一个三 三维空间，一个所谓的三维流形，所以是幸运的， 对这种几何物体的自然直觉，最终， 并没有像我们预期的那样， 这些问题已通过代数计算得到解决，以获得更高的 三维流形在三维空间中仍然令人困惑 案子 其中最著名的是著名的猜想， 彭加勒从世纪之交开始，关于3- 三维球体，而正是原来的三维 这是唯一一个还没结的案子 调查人员正在追查一名 关于三维流形的各种问题， 奇怪的距离概念，所谓的双曲度量， 但这些问题一次又一次地被证明 明确相关的情况下，流形与更熟悉的 距离的概念。