Heegaard diagrams for $5$-manifolds

TL;DR

This paper develops a new Heegaard diagram framework for 5-manifolds, enabling their representation and classification via diagrammatic moves, and applies it to analyze Gluck twists in 4-sphere cobordisms.

Contribution

It introduces a novel Heegaard diagram approach for 5-manifolds and establishes their equivalence criteria, extending diagrammatic techniques to higher dimensions.

Findings

Every smooth 5-manifold can be represented by a Heegaard diagram.

Two diagrams represent the same 5-manifold if related by specific moves.

Heegaard diagrams for cobordisms from the 4-sphere to Gluck twists are constructed.

Abstract

We introduce a version of Heegaard diagrams for $5$-dimensional cobordisms with $2$- and $3$-handles, $5$-dimensional $3$-handlebodies, and closed $5$-manifolds. We show that every such smooth $5$-manifold can be represented by a Heegaard diagram, and that two Heegaard diagrams represent diffeomorphic $5$-manifolds if and only if they are related by certain moves. As an application, we construct Heegaard diagrams for $5$-dimensional cobordisms from the standard $4$-sphere to the Gluck twists along knotted $2$-spheres. This provides several statements equivalent to the Gluck twist being diffeomorphic to the standard $4$-sphere.