---
tag: 3HDRK3QYSD
kind: example
name: Weeks manifold
msc: [57K32]
instance_of: [GHVW1Z35BS]
sources:
  - url: https://en.wikipedia.org/wiki/Weeks_manifold
same_as:
  wikidata: Q7979541
  wikipedia: Weeks manifold
---

## Example

Take the complement of the [[Whitehead link|TPGBASPKE0]] in $S^3$.
The Weeks manifold $W$ is the closed orientable $3$-manifold obtained from it by [[Dehn filling|ZA405B0Z1S]] with coefficients $(5, 1)$ and $(5, 2)$ on the two components.
It is a closed [[hyperbolic 3-manifold|GHVW1Z35BS]].
Its hyperbolic volume is approximately $0.9427$.
Its first homology group is $H_1(W; \Z) \cong \Z/5 \oplus \Z/5$.

The hyperbolic structure can be checked by computer, for example with the program SnapPea of Jeffrey Weeks.

## Remarks

Gabai, Meyerhoff and Milley proved in 2009 that $W$ has the smallest volume among all closed orientable hyperbolic $3$-manifolds.
By Mostow rigidity, its volume is a topological invariant, so this is a statement about topology.

## Explanations

### L1 human

The Weeks manifold is the smallest closed orientable hyperbolic $3$-dimensional space, measured by volume.
It is made by closing up the two tube-shaped ends of the complement of a simple link in two particular ways.
Its volume, about $0.94$, is a lower bound for the volume of every other closed orientable hyperbolic $3$-manifold.