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  "body": "\n## Statement\n\nLet $M$ be a smooth [[manifold with boundary|7DS6B2BFK8]], and let $\\partial M$ be its boundary.\nThen there is an open neighbourhood $U$ of $\\partial M$ in $M$ and a [[diffeomorphism|0KZZ70ZHXP]] $c \\colon \\partial M \\times [0, 1) \\to U$ such that $c(x, 0) = x$ for every $x \\in \\partial M$.\n\n## Remarks\n\nThe map $c$, or its image $U$, is a *collar* of $\\partial M$.\nThe idea of the proof is to choose a smooth vector field near $\\partial M$ that points into $M$ along $\\partial M$.\nSuch a field is glued from local choices with a partition of unity.\nIts flow, started at the boundary points and run for short times, gives the collar.\nExample: for the closed unit disc $D^2$, the map $c(x, t) = (1 - t/2) x$ is a collar $S^1 \\times [0, 1) \\to D^2$ of the boundary circle.\nThe statement also holds for topological manifolds with boundary, a result of Morton Brown.\n\n## Explanations\n\n### L1 human\n\nNear its boundary, every smooth manifold with boundary looks like a product of the boundary with a short interval.\nA disc near its rim looks like a thin ring, a circle times an interval.\nThis lets one glue manifolds along common boundary pieces and get a smooth manifold again.\n\n### L2 human\n\nCollars are used to glue manifolds along boundary components, for example in the composition of cobordisms and in handle attachments.\nThey show that $M$ is homotopy equivalent to its interior, and that $M$ with an outer collar $\\partial M \\times [0, 1]$ attached is diffeomorphic to $M$.\nThe collar theorem is the boundary analogue of the tubular neighbourhood theorem: the inward normal bundle of $\\partial M$ is a trivial half-line bundle.\n",
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  "name": "Collar neighbourhood theorem",
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