Mathematical Sciences: RUI: Topological Embeddings in Piecewise Linear Manifolds

9626221 Venema这是一个几何拓扑学项目。Venema目前正在研究余维二中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个同调群的哪些元素可以用拓扑嵌入的（可能是野的）2球表示的问题。这是下面这个更一般的问题的一个特例：如果一个紧的n维有边界流形具有某个封闭（n-2）流形的同伦类型，那么第二个流形是否存在一个（野生的）拓扑嵌入到第一个是同伦等价的流形中？如果流形是高度连接的呢？这个项目涉及Venema在四维空间中理解打结球体的努力。具体来说，他正在研究的问题是，不同种类的球体可以形成什么样的结。在对四维空间中球体的研究中，有三种不同的球体被证明是有用的：光滑的（具有连续变化的切向量），分段线性的（由有限数量的三角形组成）和拓扑的（由连续变形形成）。前两种类型的球已经被很好地理解了，并且有一个相当发达的理论预测，当一个从球体到空间的连续函数可以被变形成一个一对一的函数，它的像是一个光滑的或分段的线性球体。本研究项目旨在了解拓扑球体的奥秘。＊＊＊

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. ***