---
tag: AP1KY4EHMA
kind: concept
name: Handlebody
msc: [57K30, 57R65]
specializes: [7DS6B2BFK8]
sources:
  - url: https://en.wikipedia.org/wiki/Handlebody
same_as:
  wikidata: Q1143042
  wikipedia: Handlebody
---

## Summary

In 3-manifold topology, a handlebody of genus $g$ is the compact orientable $3$-manifold obtained from a $3$-ball by attaching $g$ one-handles.
It is a solid version of the closed orientable surface of [[genus|NJ787GZS1N]] $g$, which is its boundary.
It is a ball for $g = 0$, a solid torus for $g = 1$, and a solid double torus for $g = 2$.
Every closed orientable $3$-manifold is obtained by gluing two handlebodies of the same genus along their boundaries, which is a [[Heegaard splitting|EDE9919B5R]].
More generally, the word handlebody is also used for any manifold with a given [[handle decomposition|9V11N8VVTP]].

## Explanations

### L1 human

A handlebody is the solid inside of a surface with holes, like a solid doughnut or a solid pretzel.
Take a ball of clay and attach $g$ solid handles to it, like the handle of a cup; the result is a handlebody of genus $g$.
A solid doughnut is the handlebody of genus $1$, and its surface is the torus.
A thickened version of any connected graph drawn in space is also a handlebody.