# How to depict 5-dimensional manifolds

## TL;DR

This paper explores methods to visually represent 5-dimensional manifolds using lower-dimensional diagrams, enabling topological and geometric analysis despite human visualization limitations.

## Contribution

It introduces a novel approach to depict 5-manifolds through 3D and 2D diagrams, extending existing techniques for lower-dimensional manifolds.

## Key findings

- Representation of 5-manifolds via 3D diagrams
- Use of open books and contact geometry for visualization
- Application of diagrams to solve topological questions

## Abstract

We usually think of 2-dimensional manifolds as surfaces embedded in Euclidean 3-space. Since humans cannot visualise Euclidean spaces of higher dimensions, it appears to be impossible to give pictorial representations of higher-dimensional manifolds. However, one can in fact encode the topology of a surface in a 1-dimensional picture. By analogy, one can draw 2-dimensional pictures of 3-manifolds (Heegaard diagrams), and 3-dimensional pictures of 4-manifolds (Kirby diagrams). With the help of open books one can likewise represent at least some 5-manifolds by 3-dimensional diagrams, and contact geometry can be used to reduce these to drawings in the 2-plane. In this paper, I shall explain how to draw such pictures and how to use them for answering topological and geometric questions. The work on 5-manifolds is joint with Fan Ding and Otto van Koert.