Diagrams for Relative Trisections

TL;DR

This paper introduces a diagrammatic method for representing trisections of 4-manifolds with boundary, extending the analogy of Heegaard diagrams to four dimensions and providing foundational examples.

Contribution

It establishes a correspondence between 4-manifold trisections and diagrammatic representations involving surfaces and curves, advancing the understanding of 4-manifold decompositions.

Findings

Provides a diagrammatic framework for 4-manifold trisections

Establishes a correspondence analogous to sutured Heegaard diagrams

Includes foundational examples illustrating the method

Abstract

We establish a correspondence between trisections of smooth, compact, oriented $4$--manifolds with connected boundary and diagrams describing these trisected $4$--manifolds. Such a diagram comes in the form of a compact, oriented surface with boundary together with three tuples of simple closed curves, with possibly fewer curves than the genus of the surface, satisfying a pairwise standardness condition. This should be thought of as the $4$--dimensional analog of a sutured Heegaard diagram for a sutured $3$--manifold. We also give many foundational examples.