Topology of Four-Manifolds

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Proposal: DMS-0072722Abstract:This project is centered around the classification theory of topological 4-dimensional manifolds. One goal is to find sufficient algebraic conditions for splitting of a 4-manifold as a connected sum, up to ans-cobordism, in terms of the fundamental group and the intersection pairing. The main techniques for classification of manifolds, surgery and s-cobordism conjectures, remain open for 4-manifolds with large fundamental groups. This project will focus on a new approach to the problem, searching for involutions on certaincanonical examples, and for possible obstructions from gauge theory.The classification of possible large-scale structures in dimension 4,which locally look like the 4-dimensional space-time, turned out to bea much more subtle problem than in higher dimensions. This area ofresearch lies at the intersection of topology, geometry, analysis and physics. This project is aimed at a better understanding of 4-dimensional objects with large fundamental groups, which contain many loops that cannot be contracted. In particular, one goal is to analyze when a given object may be decomposed into more elementary building blocks.