Trisections Diagrams for Smooth 4-manifolds

Abstract:
A trisection of a smooth, compact, oriented 4-manifold X is a decomposition of X into three diffeomorphic codimension 0 submanifolds which have certain nice intersection properties. This decomposition, which is a 4-dimensional analog of Heegaard splittings of 3-manifolds, is known to exists for all smooth, compact 4-manifolds. There are many similarities between the 3 and 4-dimensional theories. In particular, every 4-manifold (as above) can be uniquely described by a closed surface together with three collections of simple, closed, curves; much like a Heegaard diagram. In this talk, we will provide a one-to-one correspondence between trisection diagrams and trisected 4-manifolds. If time permits, we will discuss how to glue (relative) trisection diagrams together in a way that corresponds to gluing (relatively) trisected 4-manifolds together along diffeomorphic boundary components.