# From Heegaard splittings to trisections; porting 3-dimensional ideas to dimension 4

## TL;DR

This paper explores the analogy between Heegaard splittings of 3-manifolds and trisections of 4-manifolds, clarifying their definitions, perspectives, and extensions to manifolds with boundary and embedded submanifolds.

## Contribution

It develops a comprehensive framework connecting 3D and 4D manifold decompositions, emphasizing the Morse theoretic and diagrammatic perspectives, and extends these ideas to relative settings.

## Key findings

- Clarifies the analogy between Heegaard splittings and trisections.
- Provides detailed descriptions in Morse theoretic and diagrammatic terms.
- Extends the theory to 4-manifolds with boundary and embedded submanifolds.

## Abstract

These notes summarize and expand on a mini-course given at CIRM in February 2018 as part of Winter Braids VIII. We somewhat obsessively develop the slogan `Trisections are to 4-manifolds as Heegaard splittings are to 3-manifolds', focusing on and clarifying the distinction between three ways of thinking of things: the basic definitions as decompositions of manifolds, the Morse theoretic perspective and descriptions in terms of diagrams. We also lay out these themes in two important relative settings: 4-manifolds with boundary and 4-manifolds with embedded 2-dimensional submanifolds.