Topology and Geometry at the Interface of Dimensions 3 and 4

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Understanding 4-dimensional shapes called smooth 4-manifolds is a fundamental challenge in the mathematical fields of geometry and topology and is intimately related to theoretical physics through the framework of gauge theory. Surprising results from the 20th century indicate that classification of 4-manifolds is qualitatively much different, and more difficult, than corresponding questions in both higher and lower dimensions. A fruitful approach to studying 4-manifolds is to cut them into simpler pieces along 3-dimensional manifolds, so that effective tools and techniques available from the comparatively simpler study of 3-manifolds can be brought to bear on the 4-dimensional realm. This research project aims to deepen understanding of smooth 4-dimensional manifolds using this approach, employing tools called "Floer homologies," which stem from gauge theory and mathematical physics. The project will address questions that lie at the border of 3- and 4-dimensions. For instance, it will explore algebraic structures called homology cobordism groups, which allow 3-manifolds to be added and subtracted by considering certain 4-manifolds that interpolate between them like frames of a movie. The project will address several fundamental questions on these structures and refine them to account for symmetries of the spaces involved. The project will involve graduate students in the research.This project studies topological and geometric objects in low dimensions, with the aid of tools from gauge theory and symplectic geometry. Specific goals include the development of techniques for studying smooth equivariant homology cobordism groups, application of these techniques to long-standing questions on knot concordance, and characterization of knots that bound complex curves in Stein domains. The project will also create resources for students and researchers seeking to learn the tools employed in pursuit of these goals. Specific objectives in this direction include the completion and dissemination of an informal introductory text on Heegaard Floer homology. The project draws on methods from homological algebra, gauge theory, knot theory, contact geometry, and the theory of pseudo-holomorphic curves in symplectic 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.