Surfaces in 4-manifolds and modified surgery theory

Topology is the study of spaces up to continuous deformation. Spaces of dimensions one, two and three are well understood, as are spaces in dimension five and greater. In dimension four however, much remains unknown. The PI plans to use techniques that have been successful in high dimensional topology to improve the understanding of four-dimensional spaces. In addition to the research aspects of the project, the PI will organize conferences, write lecture notes for his classes, maintain a conference listing website, write research-level surveys and mentor junior researchers. The project focuses on topological 4-dimensional manifolds and lies at the intersection between low dimensional topology and high dimensional surgery theory. Topology in dimension 4 has two distinct flavors according to the category involved: smooth or topological. While smooth 4-manifolds remain very mysterious, the topological category is amenable to broad classifications. The PI’s goals are to classify topological 4-manifolds with a fixed boundary and good fundamental group as well as to classify locally flat surfaces in 4-manifolds with a fixed knot group. In particular, the PI plans to prove the non-orientable unknotting conjecture: if a non-orientable locally flat surface in the 4-sphere has knot group of order two, then it is topologically unknotted. 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.