Some Problems in Geometric Topology

建议：DMS-9971296 PI：Steve FerryAbstract：我们的主要工作将是使用受控手术理论的方法来研究Bryant，Mio，Weinberger和作者发现的奇异同调流形的几何性质。在这一领域最紧迫的问题涉及的存在性奇异的同源流形是均匀的和存在性奇异的同源流形有收缩的普遍覆盖。构造一个奇异的齐次同调流形将是著名的Bing-Borsuk猜想的一个反例，而一个奇异的非球面同调流形可以给积分Novikov猜想或著名的Poincare对偶群是某个非球面流形的基本群的猜想一个反例.我们还计划用控制拓扑的方法来研究Gromov-Lawson猜想，并研究了余维2中高维野生拓扑嵌入的分类。

Proposal: DMS-9971296PI: Steve FerryAbstract: Our main effort will go into using the methods of controlled surgery theory to study of the geometric properties of the exotic homology manifolds discovered by Bryant, Mio, Weinberger, and the author. The most pressing questions in this area concern the existence of exotic homology manifolds which are homogeneous and the existence of exotic homology manifolds which have contractible universal covers. Construction of an exotic homogeneous homology manifold would give a counterexample to the well-known Bing-Borsuk conjecture, while an exotic aspherical homology manifold would give a counterexample to either the integral Novikov conjecture or the well-known conjecture that every Poincare duality group is the fundamental group of some aspherical manifold.We also plan to use methods from controlled topology to study the Gromov-Lawson conjecture and to study the classification of high-dimensional wild topological embeddings in codimension two.