Three and Four-Manifolds

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Proposal: DMS-9975171PI: Robion KirbyAbstract: This proposal concerns several problems in the smooth topology of fourdimensional manifolds. One is to find a topological definition of theSeiberg-Witten invariants, perhaps by exploiting almost complexstructures on 4-manifolds which are "tight" in some sense other thanbeing symplectic. Second is to find a natural construction of a4-manifold which has as boundary the connected sum of three copies(with alternating orientations) of a given closed 3-manifold. Thirdis (with Paul Melvin) to finish work on the handlebody structure ofcertain complex elliptic surfaces.Manifolds of dimension four are the least understood of alldimensions, yet they occur frequently, for example, in relation to thethree spatial and one temporal dimensions of physical space-time. Afundamental problem is to classify these manifolds, where classifymeans to understand them in terms of relatively simple invariantswhich can be computed. The Seiberg-Witten invariants which werediscovered in 1994 by mathematical physicists are the best current setof invariants, and much work needs to be done in understanding andextending them. One technique is handlebody theory which involves asimple way of describing a manifold in terms of simple buildingblocks.