On 4-dimensional 3-handle attachments

TL;DR

This paper develops methods to explicitly include 3-handle attachments in Kirby diagrams for 4-manifolds, extending classical calculus and providing criteria for their unique identification based on homology.

Contribution

It introduces moves involving 3-handles to extend Kirby calculus and establishes a homological criterion for unique 3-handle attachments in 4-manifolds.

Findings

Extended Kirby calculus with 3-handle moves

Homological criterion for 3-handle uniqueness

Explicit tools for including 3-handles in diagrams

Abstract

Kirby diagrams for smooth four-dimensional manifolds typically depict only the 1- and 2-handles, omitting the 3-handles. In this work, we undertake a study of 3-handle attachments and provide tools to explicitly include them in handle diagrams. We show a set of moves involving 3-handles to extend the classical Kirby calculus. Under the condition that the number of 3-handles equals the rank of the spherical part of the specific boundary's second homology group, we establish a homological criterion that identifies a geometric basis of disjoint embedded spheres in the boundary corresponding to 3-handle attachments, yielding a uniqueness theorem for 3-handle attachments.