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

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.

💡 Research Summary

The paper surveys and expands the relationships among three combinatorial representations of piecewise‑linear (PL) 4‑manifolds: Kirby diagrams, trisections, and (n + 1)‑colored graphs (often called gems). After a concise review of each theory, the authors present new results that deepen the interplay between them and outline several open problems.

Kirby diagrams encode a 4‑manifold by a framed link in S³ together with dotted components that represent 1‑handles. When the boundary is a connected sum of copies of S¹ × S², the 3‑ and 4‑handles are uniquely attached, so the diagram essentially determines a closed 4‑manifold. Trisections, introduced by Gay and Kirby, split a closed 4‑manifold into three 4‑dimensional handlebodies H₀, H₁, H₂ whose pairwise intersections are 3‑dimensional handlebodies of the same genus g, and whose triple intersection is a closed surface Σ. A (g; k₀, k₁, k₂)‑trisection diagram consists of Σ together with three families of curves (α, β, γ) that give Heegaard diagrams for the three 3‑handlebodies.

Gems are regular (n + 1)‑colored multigraphs whose colors lie in {0,…,n}. The residues obtained by deleting all edges of a given color correspond to the links of vertices in a dual colored triangulation. When every residue of a given color is a sphere, the graph represents a closed n‑manifold; otherwise it represents a singular manifold whose singularities correspond to non‑spherical residues. In dimension four, a 5‑colored graph encodes a PL 4‑manifold, and a finite set of combinatorial moves (dipole cancellations/additions, ρ‑pair switchings) preserve the underlying PL homeomorphism type.

The first original contribution (Proposition 9) extends the known Kirby‑to‑gem algorithm to the case where the handle decomposition contains 3‑handles. Starting from a gem that represents the sub‑manifold consisting only of 0‑, 1‑, and 2‑handles, the authors show how to introduce appropriate ρ‑pairs and dipoles so that the resulting graph encodes the full closed 4‑manifold, including the 3‑handles. This provides a purely combinatorial way to construct gems for exotic 4‑manifolds whose standard descriptions require 3‑handles.

The second set of results (Section 3.2, Proposition 21) deals with the reverse direction: given a gem that admits a “gem‑induced trisection” (i.e., the subgraph formed by colors {0,1,2} yields three genus‑g Heegaard splittings), one can read off a trisection diagram directly. The three families of curves α, β, γ are obtained by projecting the edges of colors 0, 1, 2 onto the central surface Σ defined by the residues of the remaining colors. This mirrors the classical correspondence in dimension three between gems and Heegaard diagrams, now lifted to dimension four.

Combining the two procedures yields a systematic method to pass from a Kirby diagram to a trisection diagram via gems. Propositions 22, 23, and 24 give explicit constructions for closed manifolds, for simply‑connected compact manifolds with connected boundary, and for the corresponding Kirby diagrams. The authors illustrate the method with concrete examples such as CP² # (–CP²) and certain connected sums of S² × S², showing that the resulting trisection diagrams have minimal or near‑minimal genus.

Beyond the technical constructions, the paper emphasizes algorithmic aspects: the moves involved are elementary and can be implemented on a computer, suggesting the possibility of automated conversion pipelines and databases of 4‑manifolds expressed in any of the three languages. The authors also discuss how the new gem‑to‑trisection correspondence may help to study PL invariants, to generate triangulations of exotic manifolds, and to explore the landscape of possible trisection genera.

Section 4 lists several open problems that point to future research directions. Notably, it asks whether every simply‑connected closed 4‑manifold admits a “balanced” trisection (all three 4‑handlebodies are 4‑disks), whether the minimal trisection genus can be efficiently read from a Kirby diagram, and how the combinatorial complexity of dipole/ρ‑pair operations affects the genus of the resulting trisection. Questions about computational complexity and practical implementation are also raised.

In summary, the paper provides a comprehensive bridge between three major combinatorial frameworks for PL 4‑manifolds, introduces new tools for handling 3‑handles within the gem setting, and establishes a direct route from Kirby diagrams to trisection diagrams. These contributions enrich the toolbox available to low‑dimensional topologists and open several promising avenues for both theoretical and computational investigations.