---
tag: 7DP7H8M5B2
kind: concept
name: Haken manifold
msc: [57K30]
specializes: [R20TB2KEQQ]
sources:
  - url: https://en.wikipedia.org/wiki/Haken_manifold
same_as:
  wikidata: Q2589609
  wikipedia: Haken manifold
---

## Summary

A Haken manifold is a compact irreducible [[3-manifold|R20TB2KEQQ]] that contains a two-sided [[incompressible surface|6N9GW98MR6]].
Such a manifold can be cut along incompressible surfaces again and again until only balls remain, a process called a Haken hierarchy.
This makes Haken manifolds the class where induction arguments work.
Many results, such as Waldhausen's topological rigidity and Thurston's hyperbolization, were first proved for them.

## Explanations

### L1 human

A Haken manifold is a $3$-dimensional space that is big enough to contain an essential surface.
Cutting along that surface gives a simpler space, and one can keep cutting until only solid balls are left.
This lets one prove facts about the whole space step by step, from the balls back up.
The $3$-torus, a cube with opposite faces glued, is Haken: a torus parallel to two opposite faces is an incompressible surface.
The $3$-sphere is not Haken, because every closed surface in it of positive genus can be compressed.