Trisecting 4-manifolds

TL;DR

This paper introduces a new way to decompose smooth, closed 4-manifolds into three standard pieces, establishing a unique structure similar to Heegaard splittings in 3-manifold topology.

Contribution

It defines the concept of trisections for 4-manifolds, proving their existence and uniqueness up to stabilization, and relates them to Morse 2-functions.

Findings

Any smooth, closed, oriented 4-manifold admits a trisection.

Trisections are unique up to a stabilization operation.

Trisections generalize Heegaard splittings to 4-manifolds.

Abstract

We show that any smooth, closed, oriented, connected 4--manifold can be trisected into three copies of $\natural^k (S^1 \times B^3)$, 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 $B^1$.