Abstract

Over the last forty years, most progress in four-dimensional topology came from gauge theory and related invariants. Khovanov homology is an invariant of knots in R^3 of a different kind: its construction is combinatorial, and connected to ideas from representation theory. There is hope that it can tell us more about smooth 4-manifolds; for example, Freedman, Gompf, Morrison and Walker suggested a strategy to disprove the 4D Poincare conjecture using Rasmussen’s invariant from Khovanov homology. It is yet unclear whether their strategy can work. I will explain a new attempt to pursue it (joint work with Lisa Piccirillo) and some of the challenges we encountered. I will also review other topological applications of Khovanov homology, with regard to smoothly embedded surfaces in 4-manifolds.