Surfaces and 4-Manifolds

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A major part of this research project concerns the validity of the 4-dimensional topological surgery conjecture for "large" fundamental groups. The PI plans to use his recently developed theory of link groups of 4-manifolds to formulate an obstruction in the context of the A-B slice problem, a reformulation of the surgery conjecture. Besides developing new invariants of 4-manifolds, this program will clarify the low-dimensional nature of topological 4-manifolds. Another part of the project concerns the asymptotic behavior of quantum representations of mapping class groups. The goal, in particular, is to use the representations that are provided by (2+1) dimensional TQFTs to investigate the rigidity properties of mapping class groups.This project fits in the general framework of classifying the possible large scale shapes of objects that locally look like the usual Euclidean space. The classification of three- and four-dimensional shapes is a particularly important and challenging problem. The results to date on topological classification in dimension four have paralleled the developments in higher dimension. This research project will help to clarify whether the analogy with higher dimensions works in general, for shapes with "large" fundamental groups.