# Trisections of 5-manifolds

## TL;DR

This paper extends the concept of trisections from 4-manifolds to 5-manifolds, proving that every smooth, oriented, compact 5-manifold admits a compatible smooth trisection.

## Contribution

It introduces the existence of smooth trisections for all 5-manifolds, generalizing previous work from 4-manifolds and boundary-compatible decompositions.

## Key findings

- Every smooth, oriented, compact 5-manifold admits a smooth trisection.
- Trisections can be made compatible with any given boundary trisection.
- The work generalizes the concept of multisections to five dimensions.

## Abstract

Gay and Kirby introduced the notion of a trisection of a smooth 4-manifold, which is a decomposition of the 4-manifold into three elementary pieces. Rubinstein and Tillmann later extended this idea to construct multisections of piecewise-linear (PL) manifolds in all dimensions. Given a PL manifold $Y$ of dimension $n$, this is a decomposition of $Y$ into $\lfloor \frac{n}{2} \rfloor + 1$ PL submanifolds. We show that every smooth, oriented, compact 5-manifold admits a smooth trisection compatible with any desired trisection of its boundary.