Mathematical Sciences: Topology of Smooth 4-Manifolds

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9626204 Akbulut This project has two parts, (a)topology of smooth 4-manifolds and (b)topology of real algebraic sets. In (a) the investigator intends to construct some new 4-manifolds (such as manifolds with b+ even, and a product of spheres) by using the new Seiberg-Witten technology and by some new algebraic topological techniques (torsion invariants for 4-manifolds). He would also like to use symplectic/contact structures to get a better structure theorem for smooth manifolds (i.e., generalized "cork" structures). In (b) he intends to continue with the "topological classification program" previously initiated jointly with H.King. He will also try to construct submanifolds of Euclidean space that cannot be isotopic to a nonsingular algebraic subset (i.e., strongly transcendental manifolds). The investigator would like to construct some new exotic 4-dimensional manifolds (generalizations of 4-dimensional spaces) that have not been detected by the current techniques. He also would like to classify "algebraic spaces" (spaces that are described by systems of polynomial equations). The mysteries of 4-dimensional geometry and topology have been recognized as of more than purely theoretical interest ever since the acceptance of time as a fourth physical dimension, e.g., in Einstein's theory of special relativity in the early years of this century. ***