Kirby diagrams, trisections and gems of PL 4-manifolds: relationships, results and open problems

TL;DR

This work synthesizes three complementary representations of PL $4$-manifolds—Kirby diagrams, trisections, and edge-colored graphs (gems)—and develops explicit, algorithmic bridges among them. It presents concrete procedures to pass from Kirby diagrams to gems and from gems to (gem-induced and indirect) trisections, deriving practical bounds for gem-invariants such as the regular genus $ ho$ and gem-complexity $k$, and providing genus estimates for trisections in terms of diagrammatic data (e.g., crossing numbers and chessboard counts). Key contributions include a new result connecting Kirby diagrams to closed manifolds via $3$-handles, a detailed construction of gem-induced trisections for manifolds with empty or connected boundary, and two indirect approaches (boundary extension and stabilizations) to obtain trisections for closed manifolds from gems. The framework enables algorithmic generation of trisection diagrams from Kirby data, furnishes tools to compare PL-structures, and points to open problems and potential applications in classifying and triangulating exotic $4$-manifolds.

Abstract

We review the main achievements regarding the interactions between gem theory (which makes use of edge-colored graphs to represent PL-manifolds of arbitrary dimension) and both the classical representation of PL 4-manifolds via Kirby diagrams and the more recent one via trisections. Original results also appear (in particular, about gems representing closed 4-manifolds which need 3-handles in their handle decomposition, as well as about trisection diagrams), together with open problems and further possible applications to the study of compact PL 4-manifolds.

Figures (24)

• Figure 1: subgraph corresponding to a crossing
• Figure 2: subgraphs corresponding to a positive (left) or negative (right) curl
• Figure 3: A Kirby diagram of $\mathbb S^2\times\mathbb D^2$ and a gem of its boundary
• Figure 4: The attachment of 4-colored edges among the vertices of a quadricolor
• Figure 5: The 5-colored graph $\Gamma(L,d)$ obtained from the framed link $({K_T},+1)$, $K_T$ being the trefoil knot

Theorems & Definitions (42)

• Theorem 1
• Definition 1
• Definition 2
• Definition 3
• Remark 1
• Theorem 2
• Definition 4
• Proposition 3
• Definition 5
• Proposition 4