Manifold Topology and Applications to Geometry

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ABSTRACTDMS-0602298 F.Thomas Farrell The goal of this project is two pronged. One direction is the basic problem of classifying homotopically equivalent manifolds up to homeomorphism. Surgery theory was developed for this purpose. But in order to make this theory effective for manifolds with a given fundamental group, it is necessary to calculate the algebraic L- and K-groups of its its integral group ring which occur in the surgery exact sequence. Direct algebraic methods have generally provenunsuccessful for this. The case of finite fundamental groups being a major exception. The techniques to be used instead will come from differential geometry, controlled topology, dynamical systems and Lie group theory. The results obtained so far using these methods have been quite encouraging. The second direction of the project is to find applications of these manifold classification results to geometry. One area that looks promising is to understand the topology of the space of all negatively curved Riemannian metrics on a smooth manifold and its quotient moduli space. There is the related following question. If one smooth structure on a closed manifold supports a negatively curved Riemannian metric, then does every other smooth structure support such a metric? And a possible application to affine geometry is to the problem of whether homeomorphic complete closed affine flat manifolds are necessarily diffeomorphic. Manifolds are geometric objects which locally resemble the space of Euclidean geometry but are usually quite different globally. For example the surface of the sphere locally resembles the plane; but is only finite in extent. Being finite in extent is what is meant by a closed manifold. A smooth manifold is one without corners or edges. For example the surface of the sphere is smooth but the surface of a cube, although a manifold, is not smooth. However these two manifolds are homeomorphic; i.e. it is easy to construct a continuously varying bijective correspondence between the points of these two surfaces. When the distance between points on a manifold is also considered, we are now talking about Riemannian manifolds and distance preserving correspondences are called isometries. For example the egg and the ball represent two different positively curved Riemannian metrics on the surface of the sphere. And these two metrics even represent different points in the moduli space of such metrics since the ball is uniformly round while the egg is not. However if only the notion of straight line is being considered (i.e. geodesic ) we are talking about affine manifolds and affine equivalences. Distance preserving correspondences take straight lines to straight lines but not necessarily vice versa.