Controlled Surgery

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This proposal concerns two topics in controlled surgery. The first is a homotoy-theoretic approach to controlled surgery on Poincare spaces of dimension 4 and greater. This provides input for a limit construction of homology manifolds. The principal new conclusion would be construction of exotic homology manifolds in dimension 4 (the Bryant-Ferry-Mio-Weinberger construction works in 6 and above) and, via a known resolution theorem, a proof of the 4-dimensional topological surgery conjecture. The second topic is a variation on surgery groups making use of controlled K-theory. This should have better formal properties, for instance should more often satisfy the Farrell-Jones isomorphism conjecture.This proposal concerns controlled surgery on Poincare spaces. Poincare spaces have global homological duality similar to that of manifolds. Given a map to a metric space one can measure the "size" of the duality structure: how far one must go to find dual cycles. Manifolds have duality with size 0 because duality follows from the local structure. Poincare spaces generally have no constraints, so the size is the diameter of the control space. The key part of the proposal is a size-reducing process: given a Poincare space with duality of size epsilon, find an equivalent Poincare space with much smaller size control. The approach proposed should work for spaces of dimension 4 and above. In this case it would provide input to an elaborate but already established argument to give a proof of the 4-dimensional topological surgery conjecture. This conjecture is the major unresolved problem in topology in dimension 4.