Some Problems in Geometric Topology

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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.