Multisections of manifolds

Abstract: Dave Gay and Rob Kirby recently introduced trisections of smooth 4-manifolds arising from their study of broken Lefschetz fibrations and Morse 2-functions. Dave asked us if this could be established using triangulations. We have done this and extended the theory to all dimensions. The idea is to split a 2k- or (2k+1)-manifold into k handlebodies, such that intersections of the handlebodies have special properties. The splitting can be viewed as mapping the manifold into a k-simplex and pulling back a decomposition into dual cubes. I'll outline the construction, give some applications and conclude with open questions. This is joint work with Hyam Rubinstein.