Mathematical Sciences: RUI: Topological Embeddings in Piecewise Linear Manifolds

9626221 Venema这是一个几何拓扑学的项目。Venema目前正在研究余维2中的拓扑嵌入的存在性问题。例如，他正在研究确定单连通四维流形的第二同调群中的哪些元素可以由拓扑嵌入(可能是野生的)2-球面表示。这是以下问题的特例：如果一个有边界的紧致n维流形具有某个闭(n-2)-流形的同伦型，那么第二个流形是否存在一个(野)拓扑嵌入到第一个同伦等价的流形中？如果这些集合管高度相连呢？这个项目涉及Venema在4维空间中理解多节球体的努力。具体地说，他正在研究不同类型的球体可以形成什么样的结的问题。在四维空间中球体的研究中，三种不同类型的球体已被证明是有用的：光滑的(具有连续变化的切线向量)，分段线性的(由有限个三角形组成)，以及拓扑性的(通过连续变形形成的)。前两种类型的球是相当好的理解，并且有一个相当发达的理论，它预测从球到空间的连续函数何时可以变形为像是光滑或分段线性球的一对一函数。这项研究项目旨在了解拓扑球的奥秘。***

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. ***