Mathematical Sciences: Topological Versus Smooth Rigidity

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A closed aspherical manifold is a compact connected manifold whose universal cover is contractible. Let M and N be closed aspherical manifolds, and let F be an isomorphism of their fundamental groups. Borel conjectured that F is always induced by a homeomorphism f of the manifolds. The investigator, in collaboration with L. E. Jones of SUNY-Stony Brook, intends to try to settle this conjecture. Partial results by the investigator and Jones were the main results of the previous N.S.F. grant (DMS-8801312). The project also relates to problems in differential geometry and algebraic K-theory; e.g., does Wh(G) vanish for every torsion-free group G? Manifolds are locally Euclidean spaces, which makes them very natural objects of study. For example, the physical world in which we live has three space dimensions and (in some contexts) a time dimension, making it either a three-dimensional or four-dimensional manifold. Thus the instant project to understand better the structure of manifolds might even have cosmological significance.