The 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 posed by Borel is to determine whether closed aspherical manifolds with isomorphic fundamental groups are homeomorphic. This problem has led to two families of conjectures of a functorial nature: the Isomorphism and the Fibered Isomorphism Conjectures. These conjectures 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; determining whether a smooth manifold homeomorphic to a closed negatively curved manifold supports a negatively curved Riemannian metric; and seeing if complete compact affine flat manifolds with isomorphic fundamental groups are diffeomorphic.
Manifolds are geometric objects which locally resemble the space of Euclidean geometry; but could be quite different globally. Roughly speaking a manifold M is (strongly) aspherical if this local resemblance is determined globally by a continuous correspondence p from the points in Euclidean space E to the points in M. And M is closed if this correspondence 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 are the circle and the surface of the doughnut. In the case of the circle, the correspondence 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 an integral 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.
