The Structure of Smooth 4-Manifolds

University Of California-Irvine, Irvine CA

Investigators

Abstract

Despite spectacular advances in defining invariants for simply-connected smooth and symplectic 4-dimensional manifolds and the discovery of important qualitative features about these manifolds, we seem to be retreating from any hope to classify simply-connectedsmooth or symplectic 4-dimensional manifolds. The subject is rich in examples that demonstrate a wide variety of disparate phenomena. Yet it is precisely this richness which gives us little hope to even conjecture a classification scheme. The most successful attempts have associated to each 4-dimensional manifold the solution space to complex systems of equations that arise in particle physics: the Yang-Mills equations and the monopole equations of Seiberg and Witten. These solution spaces are useful in distinguishing cunningly constructed 4-dimensional manifolds. This proposal assembles all known techniques in a manner that provides approachable questions that further explore a potential classification scheme. The first goal is to understand why smooth structures on 4-manifolds are sensitive to local topological change. The second step is to assemble the local topology into new invariants that will distinguish those manifolds with the same Seiberg-Witten invariants already constructed by the principal investigator. The core of this project is to determine a sequence of operations that relate homeomorphic smooth 4-manifolds. Excitement has been generated by the idea that the puniest of all forces, gravity, may in fact be a strong as nature's other three fundamental forces: the strong force which binds protons and neutrons together in atomic nuclei; the weak force which governs radioactive decay; and the forces that govern electricity and magnetism. The perceived mismatch between these three forces and gravity creates a theoretical nightmare; it's the principle reason we have yet to find a grand unified theory. However, it has recently been hypothesized that this weakness is a mirage; the force of gravity only appears weak because its force is diluted in our own universe and most of gravity's force radiates out into extra dimensions. All other forces remain trapped in our 3-dimensional world, while gravity is free to roam other dimensions. With this hypothesis, there could be other worlds that are parallel to our own; they all neatly stack up, each oblivious of the other, with gravity the only force that moves between them. New mathematics will be generated in this project to further explore these ideas. Much of the relevant mathematics has already exposed the special nature of dimensions three and four. The passage from one universe to another could be explained by a local change known as a logarithmic transform on a null-homologous torus. This operation captures many of the features of worm holes. It is the purpose of this project to explore the conjecture that any two homeomorphic smooth 4-manifolds are related by a sequence of such transformations. At bottom, the goal of this project is to develop more systematic constructions of smooth 4-dimensional manifolds with the hope that a general picture begins to emerge that will at least suggest a classification scheme.