Trisecting 4-manifolds

We show that any smooth, closed, oriented, connected 4–manifold can be trisected into three copies of ${\mathrm{♮}}^k(S^1\times B^3)$, intersecting pairwise in 3–dimensional handlebodies, with triple intersection a closed 2–dimensional surface. Such a trisection is unique up to a natural stabilization operation. This is analogous to the existence, and uniqueness up to stabilization, of Heegaard splittings of 3–manifolds. A trisection of a 4–manifold $X$ arises from a Morse 2–function $G:X\to B^2$ and the obvious trisection of $B^2$, in much the same way that a Heegaard splitting of a 3–manifold $Y$ arises from a Morse function $g:Y\to B^1$ and the obvious bisection of $B^1$.