Trisecting $4$–manifolds

Abstract

We show that any smooth, closed, oriented, connected $4$–manifold can be trisected into three copies of ${♮}^k\left(S^1\times B^3\right)$, 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 $B^1$.