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  "body": "\n## Statement\n\nLet $M$ be a [[3-manifold|R20TB2KEQQ]] with boundary $\\partial M$, and let $F \\subseteq \\partial M$ be a connected surface.\nLet $N$ be a [[normal subgroup|0CB0DX5BVV]] of the [[fundamental group|S1XK3HXHFX]] $\\pi_1(F)$.\nAssume that the kernel of the map $\\pi_1(F) \\to \\pi_1(M)$ induced by inclusion is not contained in $N$.\nThen there is an embedding $g \\colon (D^2, \\partial D^2) \\to (M, F)$ of the closed disk such that the loop $g|_{\\partial D^2}$ represents an element of $\\pi_1(F)$ that is not in $N$.\n\n## Remarks\n\nThe version with $N$ is due to Stallings; Papakyriakopoulos proved the case $N = \\set{1}$ in 1957.\nThe proof uses the same tower of double coverings as the proof of [[Dehn's lemma|FH5WSPD79F]].\nTake $N = \\set{1}$, and suppose some loop that is essential in $F$ becomes null-homotopic in $M$.\nThen some embedded essential loop in $F$ bounds an embedded disk in $M$.\nFor example, if $M$ is a solid torus $D^2 \\times S^1$ and $F$ is its boundary torus, the meridian disk $D^2 \\times \\set{1}$ is such a disk.\n\n## Explanations\n\n### L1 human\n\nIf a loop on the boundary of a $3$-dimensional space can be shrunk inside the space but not on the boundary, then there is an embedded disk that shows it.\nThe disk has its edge on the boundary and does not cross itself.\nSo an algebraic fact about loops produces a concrete geometric object.\n\n### L2 human\n\nThe loop theorem is the reason why \"compressible\" can be tested with fundamental groups.\nA two-sided surface in a $3$-manifold is incompressible exactly when its fundamental group injects into that of the manifold.\nTogether with Dehn's lemma and the sphere theorem, it is a basic step in the theory of Haken manifolds.\n",
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