2025/10/23 by Eva Horvat, Horvat, Eva, Michal Jablonowski +2
Engineering · Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Structural Analysis and Optimization #math.GT #msc:57R52 #msc:57R65
paper · pdf · doi:10.48550/arxiv.2510.20282
arxiv created 2026/07/31 · arxiv updated 2026/08/03
Kirby diagrams for smooth four-dimensional manifolds typically depict only the 1- and 2-handles, omitting the 3-handles. In this work, we investigate 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 assumptions and 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.