Gay and Kirby recently generalised Heegaard splittings of 3-manifolds to
trisections of 4-manifolds. A trisection describes a 4-dimensional manifold
as a union of three 4-dimensional handlebodies. The complexity of the
4–manifold is captured in a collection of curves on a surface, which guide
the gluing of the handelbodies. The minimal genus of such a surface is the
trisection genus of the 4-manifold.
After defining trisections and giving key examples and applications, I will describe an algorithm to compute trisections of 4-manifolds using arbitrary triangulations as input. This results in the first explicit complexity bounds for the trisection genus of a 4-manifold in terms of the number of pentachora (4-simplices) in a triangulation. This is joint work with Mark Bell, Joel Hass and Hyam Rubinstein. I will also describe joint work with Jonathan Spreer that determines the trisection genus for each of the standard simply connected PL 4-manifolds.