Mathematical Sciences: Invariants for 3- and 4-Manifolds

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9302526 Fintushel Since the 1982 work of Simon Donaldson, the principal investigator and R. Fintushel have applied gauge theory to study homology 3-spheres and smooth 4-manifolds. The current project follows in this tradition, utilizing the recent work of the principal investigator and R. Fintushel and the recent work of Taubes, the work of Morgan, Mrowka, and Ruberman, and the work of Kronheimer and Mrowka to determine those operations on a given smooth 4-manifold which preserve its homeomorphism type and alter its diffeomorphism type. Further, exotic smooth structures will be investigated for those manifolds which have no Donaldson polynomials, i.e. manifolds with Euler characteristic plus signature not divisible by 4. The major thrust of this project is to classify smooth simply-connected 4-manifolds, i.e. objects that are locally modeled on 4-dimensional space. Relativistic space-time is, of course, the best known example of such a manifold. Although the precise global structure of space-time is not known, many such manifolds can be described as solutions to systems of complex polynomials. Only recently have there been discovered examples which cannot be so described, thus dashing immediate hopes of classifying 4-manifolds. Because of these examples, there is no intelligent conjecture as to what form this classification might take. It is the fundamental purpose of the current project to investigate these and other "exotic 4-manifolds" further and to place them in a larger and less ad hoc framework. ***