  We offer a new proof that two closed oriented 4-manifolds are cobordant if
their signatures agree, in the spirit of Lickorish's proof 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 that the mapping class group of a surface is finitely presented.
