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  "body": "\n## Definition\n\nLet $M$ be a compact connected [[orientable|702SNFQG0W]] [[3-manifold|R20TB2KEQQ]].\nThe manifold $M$ is *irreducible* if every embedded $2$-sphere in $M$ bounds an embedded $3$-ball in $M$.\nThe manifold $M$ is a *Haken manifold* if:\n\n1. $M$ is irreducible, and\n2. $M$ contains a properly embedded two-sided [[incompressible surface|6N9GW98MR6]].\n\n## Remarks\n\nExample: the $3$-torus $T^3 = T^2 \\times S^1$ is irreducible, since its universal cover $\\R^3$ is irreducible by Alexander's theorem.\nThe torus $T^2 \\times \\set{1}$ is two-sided, and the inclusion $\\Z^2 \\to \\Z^3$ on fundamental groups is injective, so it is incompressible.\nHence $T^3$ is Haken.\n\nNon-example: the $3$-sphere $S^3$ is irreducible by Alexander's theorem, but it is not Haken.\nA closed surface $S \\subseteq S^3$ of positive genus has $\\pi_1(S) \\ne 1 = \\pi_1(S^3)$, so by the loop theorem it is compressible.\n\nEvery closed orientable irreducible $3$-manifold with infinite first homology group is Haken.\nFor non-orientable manifolds, irreducible is replaced by $P^2$-irreducible, which also forbids embedded two-sided projective planes.\n",
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    "name": "Haken manifold",
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