Mathematical Sciences: Knots, Framed Manifolds, and Jet Groups

数学科学：结、框架流形和射流群

• 批准号: 9103556
• 负责人: Alexandru Suciu
• 金额: $ 4.78万
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-07-01 至 1993-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9103556&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Knots; Framed; Manifolds

This research project involves three different areas of topology: knot theory, manifold theory, and homology of groups. The problems considered are handled by a variety of techniques, coming from differential topology (cutting and pasting, transversality, transformation groups), homotopy theory (homotopy groups of spheres, plus construction, S-duality), and algebra (homological algebra, combinatorial group theory, symmetric forms). Part 1 is an investigation of certain classes of higher- dimensional knots. The first class consists of knots which are not determined by their complements. Only recently has it been proved that such knots do not occur in 3-space. On the other hand, the investigator has since shown that such knots do occur in at least a quarter of all higher dimensions. He intends to find inequivalent knots with the same complement in the missing dimensions. The second class consists of certain knots whose complements have cyclic fundamental group. The classification of such knots will be pursued by means of algebraic invariants associated to their complements. The third class consists of knotted higher-genus surfaces in 4-space. That this is a much richer collection than that of ordinary knotted 2-spheres will be shown by an analysis of the peripheral structure of the knot groups. Part 2 concerns a special class of closed manifolds -- those obtained by spinning lower-dimensional ones about framed submanifolds of spheres. Using a generalized Pontrjagin-Thom construction, the investigator will carry out the homotopy classification of such manifolds in certain cases. Applications to the bordism classification of homology spheres will be given. Part 3 describes an ongoing program with S. Jekel for computing the homology of the classifying space for codimension- one real analytic singular foliations. The investigator outlines the calculation of the homology of discrete groups of jets of local diffeomorphisms, and of some related nilpotent groups. In short, the investigator will apply himself on the one hand, to some very concrete geometric problems concerning higher dimensional knots and symmetries of manifolds, and on the other hand, to enlarging the scope of algebraic tools for attacking such problems.

这个研究项目涉及三个不同的拓扑学领域：纽结理论、流形理论和群的同调。所考虑的问题是通过各种技巧来处理的，这些技巧来自于微分拓扑学(剪切和粘贴、横截性、变换群)、同伦理论(球面的同伦群、加构造、S对偶)和代数(同调代数、组合群论、对称形式)。第一部分是对某些类型的高维纽结的研究。第一类由结组成，这些结不是由它们的互补决定的。直到最近才被证明，这种结不会出现在三维空间中。另一方面，研究人员已经证明，在所有更高的维度中，至少有四分之一确实会出现这样的结。他打算在缺失的维度中找到具有相同补码的不等价纽结。第二类由某些纽结组成，这些纽结的补有循环的基本群。这种纽结的分类将通过与它们的补码相关的代数不变量来进行。第三类由4维空间中的纽结高亏格曲面组成。这是一个比普通打结的2-球体丰富得多的集合，这将通过对结群的外围结构的分析来显示。第2部分涉及一类特殊的闭流形--通过围绕球面的框架子流形旋转低维流形而得到的闭流形。利用推广的Pontrjagin-Thom结构，研究者将在某些情况下对这类流形进行同伦分类。文中将给出同调球的边界分类的应用。第三部分描述了S.Jekel正在进行的计算余维-一实解析奇异叶的分类空间的同调的程序。给出了局部微分同胚离散射流群及其相关幂零群的同调的计算方法。简而言之，研究者将一方面致力于一些非常具体的几何问题，涉及到流形的高维纽结和对称性，另一方面，扩大了解决这类问题的代数工具的范围。