Smooth Structures in Low-Dimensional Topology

Nate Dowlin , Princeton University
Fine Hall 314

Smooth 4-manifolds are without question the least understood topic in low-dimensional topology. In dimensions 5 and higher, the h-cobordism theorem handles most of the difficulties, while in dimensions 3 and below, smooth and topological manifolds coincide. Dimension 4 is special – it is the only low dimension in which the smooth Poincare conjecture is unsolved, and it is the only dimension in which Euclidean space admits exotic smooth structures. We will discuss where the wildness of four-dimensions comes from, the tools available for dealing with it, and show how knot theory allows us to construct some of the exotic smooth structures on R^4.