Mathematical Sciences: RUI: Topological Embeddings in Piecewise Linear Manifolds

数学科学：RUI：分段线性流形中的拓扑嵌入

• 批准号: 9626221
• 负责人: John Ferdinands
• 金额: $ 0.98万
• 依托单位: Calvin University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-15 至 1999-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626221&HistoricalAwards=false
• 关键词: Mathematical; Sciences; RUI; Topological; Embeddings

9626221 Venema This is a project in geometric topology. Venema is currently investigating problems involving existence of topological embeddings in codimension two. For example, he is working on the problem of determining which elements of the second homology group of a simply-connected 4-dimensional manifold can be represented by topologically embedded (possibly wild) 2-spheres. This is a special case of the following, more general, problem: If a compact n-dimensional manifold-with-boundary has the homotopy type of some closed (n-2)-manifold, then is there a (wild) topological embedding of the second manifold into the first which is a homotopy equivalence? What if the manifolds are highly connected? This project concerns Venema's efforts to understand knotted spheres in 4-dimensional space. Specifically, he is investigating the question of what sorts of knots can be formed from different kinds of spheres. In the study of spheres in 4-dimensional spaces, three different kinds of spheres have proved to be useful: those that are smooth (possess continuously varying tangent vectors), those that are piecewise linear (made up of a finite number of triangles), and those that are topological (formed by continuous deformation). Spheres of the first two types are fairly well understood, and there is a reasonably well developed theory which predicts when a continuous function from a sphere into a space can be deformed to a one-to-one function whose image is a smooth or piecewise linear sphere. This research project aims to understand the mysteries of topological spheres. ***

9626221 Venema 这是一个几何拓扑项目。 Venema 目前正在研究涉及余维二中拓扑嵌入存在的问题。 例如，他正在研究确定简单连接的 4 维流形的第二同调群的哪些元素可以由拓扑嵌入（可能是野生的）2-球体表示的问题。 这是以下更一般问题的特例：如果一个有边界的紧致 n 维流形具有某个闭合 (n-2) 流形的同伦类型，那么是否存在 (wild) 拓扑嵌入将第二个流形放入第一个流形中哪个是同伦等价？ 如果流形高度连通怎么办？ 该项目涉及 Venema 为理解 4 维空间中的打结球体所做的努力。 具体来说，他正在研究不同种类的球体可以形成什么样的结的问题。 在 4 维空间中的球体研究中，三种不同类型的球体已被证明是有用的：光滑的球体（具有连续变化的切向量）、分段线性的球体（由有限数量的三角形组成）、以及拓扑结构（通过连续变形形成）。 前两种类型的球体已经相当容易理解，并且有一个相当完善的理论，该理论预测从球体到空间的连续函数何时可以变形为其图像是平滑或分段线性的一对一函数领域。 该研究项目旨在了解拓扑球体的奥秘。 ***