Spaces of Embeddings

9971293设P和N是流形，其中P是紧的。这个项目分析了E(P，N)=从P到N的嵌入空间的同伦型。目的是建立关于嵌入空间的‘重析和’结果。多重析取的应用包括：1.当N是欧氏空间时，E(P，N)的正次同伦群的有限生成；2.在亚稳态范围外嵌入的硼主义理论障碍；3.高阶连接现象的研究.流形是具有齐性性质的空间：它们在局部上看起来像欧几里德空间。流形是数学研究的基本对象。研究它们的一种方法是试图确定一种流形何时“位于”另一种流形之中。这就产生了‘’嵌入‘’的说法。具体地说，流形P在流形N中的嵌入是从P到N的函数，使得P的任何两点都不通过该函数映射到N的同一点。从P到N的所有嵌入的集合形成了一个拓扑空间E(P，N)，称为‘’嵌入空间‘’。这个项目的目标是通过引进代数拓扑学领域的技术来更好地理解空间E(P，N)。

9971293Klein Let P and N be manifolds, where P is compact. This project analysesthe homotopy type of E(P,N) = the space of embeddings from P to N. Theaim will be to establish ``multiple disjunction'' results for spaces ofembeddings. Applications of multiple disjunction are to include: 1. the finite generation of the homotopy groups of E(P,N) in positive degrees when N is euclidean space, 2. bordism theoretic obstructions to embedding beyond the metastable range, and 3. the study of higher order linking phenomena. Manifolds are spaces having a homogeneity property: they locally``look like'' euclidean space. Manifolds are fundamental objects ofmathematical research. One approach to studying them is to try to decidewhen one manifold ``sits inside'' another one. This gives rise to thenotion of ``embedding.'' Specifically, an embedding of a manifold P ina manifold N is a function from P to N such that no two points of P aremapped via the function to the same point of N. The set of all embeddingsfrom P to N forms a topological space E(P,N), called the ``embeddingspace.'' The goal of this project is to understand better the spaceE(P,N) by importing techniques from the field of algebraic topology.***