Georgia Institute of Technology College of Sciences

This is joint work with Rob Kirby. Trisections are to 4-manifolds as Heegaard splittings are to 3-manifolds; a Heegaard splitting splits a 3-manifolds into 2 pieces each of which looks like a regular neighborhood of a bouquet of circles in R^3 (a handlebody), while a trisection splits a 4-manifold into 3 pieces of each of which looks like a regular neighborhood of a bouquet of circles in R^4. All closed, oriented 4-manifolds (resp. 3-manifolds) have trisections (resp. Heegaard splittings), and for a fixed manifold these are unique up to a natural stabilization operation. The striking parallels between the two dimensions suggest a plethora of interesting open questions, and I hope to present as many of these as I can.