9626101 Ferry The investigator plans 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, the main project is to discover to what extent the classical geometric theory of topological manifolds carries over to these newly discovered spaces. This study is closely related to the well-known Borel and Bing-Borsuk Conjectures. Other problems on the investigator's list of projects include the construction of controlled Gamma-surgery theory and its application to the study of topological embeddings in codimension two, the construction of manifold structures on acyclic Poincare duality spaces, the study of ``short'' maps defined on ``large'' Riemannian manifolds, and applications of controlled topology to the study of spaces of Riemannian manifolds. Much of topology is concerned with the study of mathematical objects called spaces. The surface of a sphere is a space, as is the surface of a donut. In the study of mathematical systems with many variables, it is common for spaces of very high or even infinite dimension to play important roles. A classical problem in topology is to find small checkable sets of axioms which characterize particular spaces. One popular set of axioms involves the notion of connectivity. Roughly speaking, a space is connected if it is all in one piece. It is locally connected if it can be chopped up into arbitrarily small connected pieces. (Be warned -- these plain English definitions are at best rough approximations to the actual mathematical definitions.) In 1978, James Cannon made an amazingly perceptive conjecture. He conjectured that if a space satisfied certain generalized connectivity axioms and had enough room in it to push certain sets apart, then it was a topological manifold -- a space which is assembled by gluing together small pieces of ordi nary Euclidean space. (Both the surface of a sphere and the surface of a donut are topological manifolds of dimension 2. The ``curved spaces'' appearing in general relativity are topological manifolds of dimension 4.) Combining work of F. Quinn and R. D. Edwards shows that Cannon's conjecture is true whenever a connected space contains even the tiniest manifold piece. This manifold piece acts as a sort of seed which determines the entire local structure of the space. Working with J. Bryant, W. Mio, and S. Weinberger, the investigator has shown that Cannon's conjecture is not true in complete generality. The main question the investigator will be studying is whether the ``seeding'' phenomenon from the Edwards-Quinn case holds in general -- whether the local structure of connected counterexamples is the same at every point. If this turns out to be true, these new spaces could become objects of interest paralleling manifolds. Cannon's axioms guarantee that from a large-scale point of view, these spaces look exceedingly like manifolds. If the seeding phenomenon holds, one imagines that there could eventually be theories speculating that we live in one of these new mathematical objects rather than in ``ordinary'' Euclidean space. ***
