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这是几何拓扑的项目。 Venema目前正在调查涉及Codimension二中存在拓扑嵌入的问题。 例如，他正在研究确定简单连接4维流形的第二个同源性组的哪个元素可以用拓扑嵌入的（可能是野生）2个spheres表示。 这是以下更普遍的问题的一种特殊情况：如果紧凑的n维歧管与边界有一些封闭的（n-2）-manifold的同型类型，那么第二种歧管是否有（野性）拓扑嵌入第一个歧管中的第一个歧管，这是同质拷贝等效的吗？ 如果歧管高度连接怎么办？ 该项目涉及Venema在4维空间中了解打结的球体的努力。 具体而言，他正在研究以不同类型的领域形成哪种结的问题。 在4维空间中的球体的研究中，事实证明，三种不同类型的球是有用的：光滑的球（具有连续变化的切线向量），那些是分段线性的（由有限的三角形数量），以及那些是拓扑的（由拓扑形成）。 前两种类型的领域已经相当理解，并且有一个相当发达的理论，可以预测何时从球体到空间的连续函数何时可以变形为一对一函数，该功能是平滑或分段线性球体的一对一函数。 该研究项目旨在了解拓扑领域的奥秘。 ***