Topology of 3 and 4-manifolds via bordered Floer homology

If a small piece is cut out of the surface of a beach ball or the surface of an inner tube, the resulting bit of rubber can be flattened out to look like a piece of the two-dimensional Euclidean plane. (Picture a sheet of paper, or the top of a table). Both surfaces are examples of two-dimensional manifolds; although they locally look like the plane, they are globally different from it and from each other. This project studies three-dimensional manifolds, which describe the possible global shapes of the universe, and four-manifolds, which describe the possible shape of space-time, using mathematical tools related to the quantum theory of fields. There are many such topological quantum field theories (TQFTs), but amazingly almost everything one knows about smooth four-manifolds comes from studying a single TQFT. The project aims to understand this theory more deeply in the hopes of understanding why this is and if there are other TQFTs like it. Project funds will support graduate students and undergraduate students mentored by the PI. Building on his extensive previous experience with undergraduate summer research and industrial placements, the PI will make special efforts to recruit and support members of groups under-represented in mathematics. From a more technical standpoint, the project will study Floer homology for 3-manifolds and its applications to topology in dimensions 3 and 4. The main focus will be on understanding the Floer homology of 3-manifolds with boundary and how it fits into an extended TQFT. Project research will focus on three related areas. The first involves extending invariants developed in the PI's previous work with Hanselman and Watson to manifolds with several torus boundary components. This will be used to study the invariants of satellite knots. The second involves showing that these invariants fit into the wider structure of a 2-3-4 TQFT. This work will draw on ideas from symplectic geometry and the theory of extended TQFT’s. The third area involves a large, but poorly studied, class of 3-manifolds known as Floer simple manifolds. The PI will study the topology of these manifolds and their relation to hyperbolic geometry and the L-space conjecture. 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.