Functions on surfaces and constructions of manifolds in dimensions three, four and five

We offer a new proof that two closed oriented 4–manifolds are cobordant if their signatures agree, in the spirit of Lickorish’s proof [6] that all closed oriented 3–manifolds bound 4–manifolds. Where Lickorish uses Heegaard splittings we use trisections. In fact we begin with a subtle recasting of Lickorish’s argument: Instead of factoring the gluing map for a Heegaard splitting as a product of Dehn twists, we encode each handlebody in a Heegaard splitting in terms of a Morse function on the surface and build the 4–manifold from a generic homotopy between the two functions. This extends up a dimension by encoding a trisection of a closed 4–manifold as a triangle (circle) of functions and constructing an associated 5– manifold from an extension to a 2–simplex (disk) of functions. This borrows ideas from Hatcher and Thurston’s proof [3] that the mapping class group of a surface is finitely presented.