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

TL;DR

This paper reviews the connections between gem theory, Kirby diagrams, and trisections in representing PL 4-manifolds, introduces new results on gems and trisection diagrams, and discusses open problems and future directions.

Contribution

It provides a comprehensive review of the relationships between different representations of PL 4-manifolds and presents new findings on gems and trisection diagrams.

Findings

New results on gems representing closed 4-manifolds with 3-handles

Original insights into trisection diagrams

Discussion of open problems and potential applications

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.