Mathematical Sciences: Some Problems in Geometric Topology

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The investigator intends to study a number of topics in controlled topology and differential geometry. The most pressing of these concern the nonresolvable homology manifolds discovered by the investigator, J. Bryant, W. Mio, and S. Weinberger. Simply put, he wishes to study these spaces and see to what extent the classical theory of manifolds carries over to this new category of spaces. This study is closely related to the well-known Borel and Bing-Borsuk Conjectures. Other problems involved include applications of controlled Gamma-surgery to the study of topological embeddings in codimension two, the Borel and integral Novikov Conjectures for certain classes of groups, and applications of controlled topology to differential geometry. There are a number of potential applications for this work. The new spaces being studied have the global properties of manifolds modelled on euclidean space, but have very different small-scale properties. Such spaces could be of interest to workers trying to reconcile large-scale and small-scale models of the universe. The Borel and Novikov Conjectures are conjectures involving the rigidity of certain classes of manifolds. Typical results say that manifolds which are "close together" in some topological sense are the same. This appears to be a naturally occurring instance of quantization in geometry and topology.