Mathematical Sciences: RUI: Topological Embeddings in Piecewise Linear Manifolds

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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. ***