Some Problems in High Dimensional Manifold Topology

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DMS-0305423F. Thomas FarrellThe goal of this project is to investigate certain problems in high dimensional manifold topology and connections with Riemannian geometry and algebraic K- and L-theories. A basic problem posedby Borel is to determine whether closed aspherical manifolds withisomorphic fundamental groups are homeomorphic. This problem has led to two families of conjectures of a functorial nature: theIsomorphism and the Fibered Isomorphism Conjectures. Theseconjectures are of central importance to much of this project. Besides Borel's other related geometric problems will be addressed.These include: a generalized Nielsen Question; replacing a map to an aspherical manifold by an approximate fibration; determiningwhether a smooth manifold homeomorphic to a closed negativelycurved manifold supports a negatively curved Riemannian metric;and seeing if complete compact affine flat manifolds with isomorphicfundamental groups are diffeomorphic.Manifolds are geometric objects which locally resemble the space ofEuclidean geometry; but could be quite different globally. Roughlyspeaking a manifold M is (strongly) aspherical if this local resemblanceis determined globally by a continuous correspondence p from the points in Euclidean space E to the points in M. And M is closed if thiscorrespondence maps some ball in E onto M. ( Caveat: The correspondence p is never one-to-one when M is closed.) The fundamental group of M consists of all the motions of E which preserve the correspondence p. Some examples of closed aspherical manifolds arethe circle and the surface of the doughnut. In the case of the circle, thecorrespondence p sends the real number x to the point (cos(x), sin(x)) and the fundamental group of the circle consists of all translations by anintegral multiple of the circumference of the circle. The Moebius band is another example of an aspherical manifold; but it is not closed. On the other hand, the sphere is a manifold which is not aspherical. A pair of closed manifolds are homeomorphic if their points can be placed in a continuously varying one-to-one correspondence. The answer to Borel's problem is well known to be Yes for aspherical manifolds of dimensions 1 or 2; but is in general unknown even for 3-dimensional manifolds.