---
tag: EDE9919B5R
kind: concept
name: Heegaard splitting
msc: [57K30]
sources:
  - url: https://en.wikipedia.org/wiki/Heegaard_splitting
same_as:
  wikidata: Q5697916
  wikipedia: Heegaard splitting
---

## Summary

A Heegaard splitting cuts a closed orientable [[3-manifold|R20TB2KEQQ]] along a closed surface into two [[handlebodies|AP1KY4EHMA]] of the same genus.
Every closed orientable $3$-manifold has one, so each such manifold is described by a surface and a gluing map between the boundaries of two handlebodies.
The smallest possible genus is the Heegaard genus, an invariant of the manifold.

## Explanations

### L1 human

A handlebody is a solid ball with some solid handles attached, like the inside of a pretzel.
Take two such solids with the same number of handles and glue their surfaces together.
The result is a closed $3$-dimensional space, and every closed orientable one arises in this way.
For example, the $3$-sphere is two solid balls glued along their boundary spheres, and also two solid doughnuts glued along their boundary tori.