Geometry and Topology Seminar
In honor of John Stallings' great paper, "How not to prove the Poincare conjecture", I will show how to reduce the smooth 4-dimensional Poincare conjecture to a (presumably incredibly difficult) statement in group theory. This is joint work with Aaron Abrams and Rob Kirby. We use trisections where Stallings used Heegaard splittings. The bigger picture is that we set up a bijection between closed, connected, oriented, smooth 4-manifolds and a certain class of purely algebraic structures which we call "group trisections", modulo a purely algebraic stabilization operation. So in the end smooth 4-dimensional topology is "just group theory"!