Some Problems in Topological Rigidity

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DMS-9987185F. Thomas FarrellThe goal of this project is to resolve some questions about the structure of manifolds. Manifolds are geometric objects which locally resemble the space of Euclidean geometry; but could be quite different globally. For example the surface of the earth, on a small scale, can be thought of as Euclid's plane. But on a large scale, it is more useful to model it on thesphere. One way to detect global differences is to travel long distances, return and see if something has changed. For example, if you walk around the surface of a Moebius band, you'll return on the opposite side. The collection of all travel routes (up to continuous deformation) forms an algebraic object called the Poincare group of the manifold. A basic conjecture formulated almost 50 years ago by A. Borel is that the Poincare group of a closed aspherical manifold determines this manifold up to a continuous bijective correspondence. Closed means without boundary but finite in extent. The Moebius band, for example, is not closed but the sphere is. Aspherical means that any continuous image of a sphere bounds the image of a ball. Spheres of dimension bigger than two and correspondingly higher dimensional balls must be allowed in this definition. For example, the Borel conjecture is true for the surface of a doughnut. However, the sphere itself is "clearly" not aspherical.Farrell in collaboration with L.E. Jones have verified some important cases of Borel's conjecture and will continue to work on it. One interesting question is whether there is a "best of all possible" maps giving the continuous bijection. It was once thought that the unique harmonic homotopyequivalence does this in the case both manifolds are negatively curved. Farrell, Jones, and P. Ontaneda have recently shown that this is not so in general. (Farrell,Ontaneda and M.S. Raghunathan extending this work have just produced such examples even with pairs of diffeomorphic manifolds: i.e. the harmonic map homotopic to the diffeomorphism is not univalent.) But it is still open whether this harmonic map is always cellular. Note if so this would contradict the Poincare Conjecture. Another problem to be investigated is whether any finite group of symmetries of the Poincare group lifts to a group of symmetries of the closed aspherical manifold provided the Poincare group has trivial center. The thrust of this problem is to find interesting cases where it is true. The case of surfaces is, for example, the classical Nielsen problem solved by Kerkhoff. Farrell and Jones have already obtained some other positive partial results on this problem. In their work on these geometric problems, Farrell and Jones have formulated conjectures about the structure of the algebraic K and L theories of an arbitrary group ring. They will continue their work on verifying these conjectures. Again the thrust of this effort is to give positive solutions for interesting groups