Kirby’s handle decomposition problem for 4-manifolds

Determine whether every closed simply connected 4-dimensional manifold admits a handle decomposition with no 3-handles.

Background

In discussing handle decompositions suggested by their flow-driven surgeries, the authors highlight a broader open problem from manifold topology, originally posed by Kirby, concerning the existence of handle decompositions excluding certain indices.

This question is relevant to the paper’s theme that flow-induced surgeries can imply missing handles, motivating connections between geometric flow singularities and classical topological handle theory.