Mathematical Sciences: Knots, Framed Manifolds, and Jet Groups

该研究项目涉及三个不同的领域， 拓扑学：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术来处理的， 来自差分拓扑（剪切和粘贴， 横截性，变换群），同伦理论（同伦 群的领域，加上建设，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等教育的调查- 空间结 第一类由结组成， 而不是由它们的互补性决定的。 直到最近， 证明了这样的结不发生在3-空间。 另 另一方面，调查人员已经证明，这种结确实发生， 至少有四分之一的更高维度 他打算 在缺少的部分中找到具有相同补数的不等价结 尺寸. 第二类由某些结组成， 补数有循环基本群。 的分类 这种纽结将通过代数不变量来实现 与其互补。 第三类包括 四维空间中的纽结高亏格曲面 这是一个非常 比普通的纽结2-球更丰富的收集将是 通过分析结的外围结构， 组 第2部分涉及一类特殊的闭流形--那些 通过围绕框架旋转低维值获得 球面的子流形 使用广义Pontrjagin-Thom 构造，研究者将执行同伦 在某些情况下，这类流形的分类。 应用 将给出同调球的边界分类。 第3部分描述了一个正在进行的计划与S。Jekel for 计算所述分类空间的余维的同调; 一个真实的解析奇异叶理。 调查人员概述了 离散喷流群同调的计算 局部幂零同态，以及一些相关的幂零群。 简而言之，调查人员一方面将致力于 一些非常具体的几何问题， 维结和流形的对称性，另一方面， 一方面，扩大代数工具的范围， 这样的问题。

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.