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  "body": "\n## Statement\n\nLet $n \\ge 2$.\nLet $\\Sigma_1, \\Sigma_2 \\subset \\R^n$ be disjoint locally flat embedded $(n-1)$-spheres, that is, images of [[topological embeddings|00ZG9Y9TR4]] $S^{n-1} \\times [-1, 1] \\to \\R^n$ restricted to $S^{n-1} \\times \\set{0}$.\nFor $i = 1, 2$, let $U_i$ be the bounded component of $\\R^n \\setminus \\Sigma_i$, and suppose $\\Sigma_1 \\subset U_2$.\nThen the region $\\overline{U_2} \\setminus U_1$ is [[homeomorphic|0QT00PB5G2]] to the annulus $S^{n-1} \\times [0, 1]$.\n\n## Remarks\n\nEach $\\overline{U_i}$ is a closed ball by the generalized [[Schoenflies theorem|3VX06XRVZG]].\nThis does not give the statement, because the inner ball may sit inside the outer one in a complicated way.\nFor $n \\le 3$ the theorem is classical; in dimension $3$ it rests on Moise's work on triangulations.\nFor $n \\ge 5$ it was proved by Robion Kirby in 1969, using the torus trick, and for $n = 4$ by Frank Quinn in 1982.\nAn equivalent form is the stable homeomorphism theorem: every orientation-preserving homeomorphism of $\\R^n$ is a finite composition of homeomorphisms that are the identity on some open set.\nThe local flatness hypothesis is necessary: wild spheres such as the Alexander horned sphere give counterexamples in dimension $3$.\n\n## Explanations\n\n### L1 human\n\nPut one ball inside a larger ball, both possibly deformed but with tame surfaces.\nThe theorem says that the shell between them is always a standard thick spherical shell, like the peel of an orange.\nIn the plane, this says that the ring between two nested loops is a deformed round ring.\n\n### L2 human\n\nThe annulus theorem was long called the annulus conjecture, and its proof was a turning point in the theory of topological manifolds.\nIt is needed to show that the connected sum of oriented topological manifolds is well defined.\nIt is also a step in Kirby and Siebenmann's study of triangulations and piecewise linear structures on topological manifolds.\nKirby's torus trick, introduced for this proof, became a standard tool of the subject.\n",
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