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  "body": "\n## Definition\n\nLet $M$ be a compact [[3-manifold|R20TB2KEQQ]].\nLet $S \\subseteq M$ be a compact connected [[surface|VGJJVBZ1MQ]] that is neither a $2$-sphere nor a closed $2$-disk.\nAssume that $S$ is either properly embedded in $M$ (that is, $S \\cap \\partial M = \\partial S$) or contained in $\\partial M$.\nA *compressing disk* for $S$ is an embedded closed disk $D \\subseteq M$ with $D \\cap S = \\partial D$ such that the curve $\\partial D$ does not bound a disk in $S$.\nThe surface $S$ is *incompressible* if it has no compressing disk, and *compressible* otherwise.\n\n## Remarks\n\nExample: let $M = T^2 \\times [0, 1]$ and $S = T^2 \\times \\set{1/2}$.\nLet $D$ be a disk in $M$ with $D \\cap S = \\partial D$.\nThe inclusion $S \\to M$ is a homotopy equivalence, so the loop $\\partial D$, null-homotopic in $M$, is null-homotopic in $S$.\nA null-homotopic simple closed curve in a surface bounds a disk in it, so $D$ is not a compressing disk, and $S$ is incompressible.\n\nNon-example: let $M = D^2 \\times S^1$ be a solid torus and $S = \\partial M$.\nThe meridian disk $D = D^2 \\times \\set{1}$ meets $S$ in $\\partial D$, and $\\partial D$ represents a nonzero class in $\\pi_1(S) \\cong \\Z^2$, so it bounds no disk in $S$.\nHence $D$ is a compressing disk and $S$ is compressible.\n\nIf $S$ is two-sided, the [[loop theorem|8YYJBVP68F]] shows that $S$ is incompressible if and only if the inclusion induces an injective map $\\pi_1(S) \\to \\pi_1(M)$.\nSome authors also call a $2$-sphere incompressible when it does not bound a $3$-ball in $M$.\n",
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  "name": "Incompressible surface",
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