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  "body": "\n## Statement\n\nLet $M$ be a [[3-manifold|R20TB2KEQQ]] and let $D$ be the closed $2$-disk.\nLet $f \\colon D \\to M$ be a piecewise linear map.\nIts singular set is the closure of the set of points $x \\in D$ such that $f^{-1}(f(x))$ contains a point other than $x$.\nIf the singular set of $f$ does not meet $\\partial D$, then there is a piecewise linear embedding $g \\colon D \\to M$ with $g = f$ on $\\partial D$.\n\n## Remarks\n\nDehn published a proof in 1910, and Kneser found a gap in it.\nPapakyriakopoulos gave a proof in 1957 with his tower construction.\nOne lifts $f$ through a tower of double coverings of regular neighborhoods of its image, until the singularities can be removed at the top.\nThen one pushes the embedded disk back down the tower.\nA key application is to knots: a tame [[knot|QYEAZTYGPT]] $K$ in $S^3$ is trivial if and only if its [[knot group|AQVHM3FP0D]] is infinite cyclic.\nIf the knot group is $\\Z$, the longitude of $K$ is null-homotopic in the complement, and Dehn's lemma turns a singular disk bounded by it into an embedded disk.\n\n## Explanations\n\n### L1 human\n\nSuppose a loop in a $3$-dimensional space bounds a disk that may cross itself, but only away from its edge.\nDehn's lemma says that the loop then also bounds a disk that does not cross itself at all.\nIt turns information about homotopy into information about embedded surfaces.\n\n### L2 human\n\nDehn's lemma, the loop theorem and the sphere theorem form the classical toolkit of $3$-manifold topology.\nThey all say that a map that is nontrivial up to homotopy can be replaced by an embedding.\nThey are used to show that an incompressible surface is the same as a two-sided surface whose inclusion is injective on fundamental groups.\n",
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  "name": "Dehn's lemma",
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