Mathematical Sciences: Low Dimensional Topology

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9501105 Freedman Freedman is concerned with a variety of problems in low dimensional geometric topology, investigating in dimension four, constructions leading to surgery and s-cobordism theorems, while attempting to formulate a uniquely 4-dimensional surgery obstruction in the case where the constructions appear to fail. In the theory of hyperbolic 3-manifolds, he studies the ends of non-compact 3-manifolds, using least area laminations and least length webs. A conjecture is that these ends are tame. This would imply the Ahlfors conjecture. The work on 4-manifolds leads to a "classification" of the torsion-free nilpotent quotients of 3-manifold groups. With the addition of a time dimension, the world we live in is a four-dimensional manifold, which makes it that much more intriguing that dimension four is where various anomalies in manifold theory occur. Freedman was one of the prime movers in this theory. For example, in all other dimensions there is only one differentiable structure on n-space, i.e. one way of doing calculus. But in dimension four, there are infinitely many different ways of doing calculus. Freedman's work continues to shed light on these dark corners of manifold theory as well as to explore their significance for mathematical physics. ***