Topology of 4-Manifolds, Embeddings, and Stable Homotopy Invariants of Links

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This project concerns the study of topological shapes, or manifolds, in dimensions 3 and 4. The classification of such shapes, locally modeled on the Euclidean space, in these dimensions is an important problem for several reasons. One is its relevance to physics; indeed, ideas from theoretical physics have led to new insights into the structure of spaces of these dimensions. Moreover, some of the outstanding open problems in topology concern manifolds specifically of dimension 4, and the study of such manifolds has important links with many areas of mathematics. This project is aimed at several questions in 4-manifold topology, including the classification up to continuous deformations of manifolds with large fundamental groups where many loops cannot be contracted. The project also has substantial broader impacts, aimed at training undergraduate and graduate students, and outreach activities.This project is aimed at several research directions, providing different methods and applications to a common set of problems: invariants of links and of link concordance, and classification of surfaces in 4-manifolds. One goal is to build on recent advances towards a resolution of the topological surgery conjecture, an open problem in 4-manifold topology, for free fundamental groups. This work will involve the construction of universal surgery problems, and related methods of the A-B slice problem. The project also aims to apply the techniques of the Goodwillie-Weiss embedding calculus of functors to embedding problems in 4-manifold topology, in particular to link concordance and to embedding of surfaces in 4-manifolds. Another direction of research concerns stable homotopy refinement of link homology theories, with applications to surfaces in 4-manifolds.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.