Mathematical Sciences: Studies in Geometric Topology

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The eventual goal of Quinn's program is to develop invariants of smooth compact 4-manifolds which, for example, could detect counterexamples to the smooth 4-dimensional Poincare conjecture. The first part of the program (this project) will be devoted primarily to study of a model problem, the Andrews-Curtis conjecture on 2-complexes. The invariants under consideration are "topological quantum field theories" in the sense of Atiyah and Witten. The Poincare conjecture asserts that any manifold that has certain simple topological properties in common with a topological sphere must actually be a topological sphere. It was originally stated for three-dimensional manifolds around the turn of the century and subsequently generalized to all higher dimensions. Curiously, all these generalizations have now been settled (in the affirmative), but the original case for three-dimensional spheres still resists all assaults. The foregoing remarks apply to topological manifolds. The situation is slightly different for smooth manifolds. Here both the three- and the four-dimensional versions of the Poincare conjecture remain open. What Quinn is hoping to do is to settle the four-dimensional case, using methods inspired by a mathematical formulation of quantum field theory.