Mathematical Sciences: Controlled Topology, Group Actions and Stratified Spaces

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9504759 Hughes This project applies techniques from controlled topology to solve problems about manifolds. The manifolds in question are of two types. The first are manifolds equipped with a natural map to the circle. Special properties of these manifolds have been investigated for the past three decades, but finally controlled topology (in a parametrized incarnation) reveals the possibility of a grand unified theory for all the previously investigated phenomena. The point of view of this project is to assemble all manifolds which map to a circle into one large space. The controlled topology can be used to factor this space into various geometrically and algebraically significant components. The techniques suggest that a study of the space of all manifolds which map to a fixed manifold of nonpositive curvature would lead to new phenomena. The point here is that the circle is the simplest manifold of nonpositive curvature. The second type of manifolds under consideration are those equipped with a group of symmetries. When this group acts on the manifold, there results an "orbit space" which is, in fact, a stratified space. The stratification presents a decomposition of the orbit space into manifold pieces. Since the orbit space is not itself a manifold, geometric methods cannot be applied until it is understood how the manifold pieces fit together. One of the most important aspects of the project is the uncovering of the geometric structure used to join together the manifold pieces. This structure is not only subject to classification and quantification, but can also be applied for the development of new geometric tools for studying stratified spaces. The structure on the neighborhoods of the strata is a bundle in the category of controlled topology. Thus, controlled topology becomes the vehicle for transforming and applying classical manifold techniques to study these more intricate manifolds with singularities. Manifolds are geometric objects that locally resemble balls in a suitable Euc lidean space. A sphere or a doughnut can serve as a familiar example of a low dimensional manifold. High dimensional manifolds are not nearly so esoteric and unphysical as one might at first believe, since dimensions correspond to degrees of freedom. Accordingly, solutions to ordinary and/or differential equations describing physical systems of many particles have three dimensions for each particle, and frequently relevant qualitative behavior of the system (as opposed to detailed quantitative behavior) is captured by the topology. Such properties of planetary orbits as being closed, instead of receding to infinity like those of some comets, are topological properties. Portions of this project deal with decompositions or stratifications of manifolds obtained by considering orbits and the reconstruction of the manifold from manifold pieces. ***