Heegaard diagrams for $5$-manifolds

TL;DR

This work generalizes Heegaard diagrams to 5-manifolds by defining 5D diagrams (Σ,α,β) with Σ a closed 4-manifold and α,β framed 2-links, and proves each relevant 5-manifold has such a diagram with equivalence given by isotopies, handle slides, stabilizations, and diffeomorphisms. It builds a bridge to Kirby calculus via singular banded unlink diagrams, enabling construction of Kirby diagrams for 2-surgeries from 5D diagrams and providing a framework to analyze 5D cobordisms and closed 5-manifolds through combinatorial moves. The paper then applies this machinery to Gluck twists, constructing a natural cobordism W_{X,K} from X to X_K and establishing several equivalent conditions for the Gluck twist on S^4 to be diffeomorphic to S^4, including a correspondence with twice-punctured S^2 tilde{×} S^3 and a banded unlink reformulation. These results give a concrete, diagrammatic toolkit for 5-manifold topology and offer new perspectives on Gluck twists, including cases involving spun knots, via Kirby moves and banded unlink manipulations.

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.

Figures (19)

• Figure 1: Left: A Kirby diagram of Mazur's contractible manifold $M$, which admits a handle decomposition consisting of a $0$-handle, a $1$-handle, and a $2$-handle. The $0$-framed attaching sphere $K$ of the $2$-handle intersects the belt sphere of the $1$-handle geometrically three times and algebraically once. Middle: A Heegaard diagram of $M\times B^1$, obtained from a Kirby diagram of the double of the left by adding a red meridian. Right: Another Heegaard diagram of $M\times B^1$, obtained from the middle diagram after sliding $K$ over the $0$-framed meridian to change the crossings of $K$. This diagram represents $B^5$ after cancelling a $(1,2)$-pair and a $(2,3)$-pair, followed by a first destabilization.
• Figure 2: Three types of handle slides. First row: A $1$-handle slide over a $1$-handle. Second row: A $2$-handle slide over a $1$-handle. Third row: A $2$-handle slide over a $2$-handle.
• Figure 3: Left: A cancelling $(1,2)$-pair. Right: A cancelling $(2,3)$-pair.
• Figure 4: Left: A vertex $v$ of $(L,\sigma)$. Top right: The positive resolution of $(L,\sigma)$. Middle right: A union of $L$ and a companion disk $c_v$. Bottom right: The negative resolution of $(L,\sigma)$.
• Figure 5: Left: A Heegaard diagram $(\Sigma,\alpha,\beta)$ of a $5$-dimensional cobordism from $S^4$ to a non-simply connected homology $4$-sphere, consisting of a $2$-handle and a $3$-handle that are algebraically but not geometrically cancelled. Alternatively, it can be interpreted as a Heegaard diagram of a contractible $5$-manifold with a $0$-handle, a $2$-handle, and a $3$-handle, which is not homeomorphic to $B^5$. Here, $\Sigma$ (in black) represents $S^2\times S^2$, $\alpha$ (in red) is the belt sphere of the $2$-handle representing $\{x_0\}\times S^2\subset S^2\times S^2$, and $\beta$ (in blue) is the attaching sphere of the $3$-handle representing a $2$-knot homotopic but not isotopic to $S^2\times \{y_0\}\subset S^2\times S^2$. Middle: A Kirby diagram of $\Sigma(\alpha)$, which is diffeomorphic to $S^4$. Right: A Kirby diagram of $\Sigma(\beta)$, which is diffeomorphic to the non-simply connected homology $4$-sphere.

Theorems & Definitions (104)

• Definition 1.1
• Remark 1.2
• Definition 1.3
• Remark 1.4
• Definition 1.5
• Remark 1.6
• Theorem 1.6
• Theorem 1.7: kim2025note
• Corollary 1.8
• Theorem 1.8