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  "body": "\n## Summary\n\nA Haken manifold is a compact irreducible [[3-manifold|R20TB2KEQQ]] that contains a two-sided [[incompressible surface|6N9GW98MR6]].\nSuch a manifold can be cut along incompressible surfaces again and again until only balls remain, a process called a Haken hierarchy.\nThis makes Haken manifolds the class where induction arguments work.\nMany results, such as Waldhausen's topological rigidity and Thurston's hyperbolization, were first proved for them.\n\n## Explanations\n\n### L1 human\n\nA Haken manifold is a $3$-dimensional space that is big enough to contain an essential surface.\nCutting along that surface gives a simpler space, and one can keep cutting until only solid balls are left.\nThis lets one prove facts about the whole space step by step, from the balls back up.\nThe $3$-torus, a cube with opposite faces glued, is Haken: a torus parallel to two opposite faces is an incompressible surface.\nThe $3$-sphere is not Haken, because every closed surface in it of positive genus can be compressed.\n",
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