
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 , 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 arises from a Morse 2--function and the obvious trisection of , in much the same way that a Heegaard splitting of a 3--manifold arises from a Morse function and the obvious bisection of .
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