Uniqueness of 4-manifolds described as sequences of 3-d handlebodies

TL;DR

This paper proves a uniqueness theorem for various descriptions of smooth, orientable, closed 4-manifolds, showing that different representations are related by specific moves, thus establishing a form of canonical form.

Contribution

It introduces a unifying uniqueness result for multiple descriptions of 4-manifolds, connecting Morse functions, loops in complexes, and multisections.

Findings

Any two loops of Morse functions yielding the same 4-manifold are related by a set of moves.

Uniqueness theorems hold for descriptions via the cut complex and pants complex.

The results unify different approaches to 4-manifold classification.

Abstract

Work of numerous authors has shown that any smooth, orientable, closed 4-manifold may be described as a loop of Morse functions on a surface, a loop in the cut complex, a loop in the pants complex, or as a multisection. In this paper, we prove a corresponding uniqueness theorem for each of these descriptions, so that, for example, any two loops of Morse functions on a surface yielding diffeomorphic 4-manifolds are related by a given set of moves.