Mathematical Sciences: Knots, Framed Manifolds, and Jet Groups

本研究项目涉及三个不同的拓扑学领域：结理论、流形理论和群的同调。所考虑的问题由各种技术处理，来自微分拓扑（剪切和粘贴，横向，变换群），同伦理论（球的同伦群，加上构造，s -对偶）和代数（同调代数，组合群论，对称形式）。第一部分是对某些高维节的研究。第一类由不由补数决定的结组成。直到最近才证明这样的结在三维空间中不会发生。另一方面，研究者已经证明这种结至少在四分之一的高维空间中确实存在。他打算在缺失的维度中找到具有相同补数的不相等结点。第二类由若干补具有环基群的结组成。这种结的分类将通过与其补数相关联的代数不变量来进行。第三类曲面由4空间中的结高格曲面组成。这是一个比普通结2球更丰富的集合，将通过对结群外围结构的分析来显示。第2部分讨论了一类特殊的闭流形——通过旋转关于球的框架子流形的低维流形得到的闭流形。利用广义的Pontrjagin-Thom构造，研究者将在某些情况下对这类流形进行同伦分类。给出了在同调球的界域分类中的应用。第3部分描述了与S. Jekel正在进行的计算余维一元实解析奇异叶的分类空间的同调的程序。研究人员概述了局部微分同态射流离散群的同调性的计算，以及一些相关的幂零群。简而言之，研究者一方面将自己应用于一些非常具体的几何问题，涉及高维结和流形的对称性，另一方面，扩大代数工具的范围，以解决这些问题。

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.