---
tag: CBXJVT7M6H
kind: definition
name: Handlebody
msc: [57K30, 57R65]
defines: AP1KY4EHMA
uses: [7DS6B2BFK8, 9V11N8VVTP, 702SNFQG0W, NJ787GZS1N]
sources:
  - url: https://en.wikipedia.org/wiki/Handlebody
---

## Definition

Let $g \ge 0$ be an integer.
A *handlebody of genus $g$* is a compact, connected, orientable $3$-dimensional smooth [[manifold with boundary|7DS6B2BFK8]] $V$ with the following property.
It has a [[handle decomposition|9V11N8VVTP]] with one $0$-handle, $g$ one-handles, and no other handles.

## Remarks

The [[orientability|702SNFQG0W]] condition fixes the way each $1$-handle is attached, and then a handlebody of genus $g$ is unique up to diffeomorphism.
Its boundary $\partial V$ is the closed orientable surface of [[genus|NJ787GZS1N]] $g$, and $V$ is homotopy equivalent to a wedge of $g$ circles.
Equivalently, for $g \ge 1$, $V$ is the boundary connected sum of $g$ solid tori $S^1 \times D^2$.
Also, $V$ is a closed regular neighbourhood of a connected finite graph in $\R^3$ with Euler characteristic $1 - g$.
Example: the solid torus $S^1 \times D^2$ is a handlebody of genus $1$.
It is the $3$-ball $D^3$ with a $1$-handle $D^1 \times D^2$ attached along two disjoint discs of $\partial D^3$, in an orientation-compatible way.
Non-example: the solid Klein bottle is obtained from $D^3$ by attaching one $1$-handle so that the result is non-orientable.
It has the right handles but is not orientable, so it is not a handlebody.
In higher-dimensional topology, *handlebody* often means any manifold with a handle decomposition, with no restriction on dimension or on the indices of the handles.