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  "body": "\n## Statement\n\nLet $M$ be a closed connected orientable [[3-manifold|R20TB2KEQQ]].\nThen there is a [[link|1A5WRCNTM3]] $L \\subset S^3$ such that $M$ is homeomorphic to the result of integral [[Dehn surgery|ZA405B0Z1S]] on $L$.\nMoreover, $L$ can be chosen so that each component is unknotted and each surgery coefficient is $+1$ or $-1$.\n\n## Remarks\n\nWallace and Lickorish proved the theorem independently, by different methods.\nIdea of Lickorish's proof: write $M$ as a [[Heegaard splitting|EDE9919B5R]] whose gluing map differs from that of a standard splitting of $S^3$ by a homeomorphism of the Heegaard surface.\nBy the [[Dehn–Lickorish theorem|BG3EH18KRE]], this homeomorphism is isotopic to a product of Dehn twists.\nEach Dehn twist can be realised by a $\\pm 1$ surgery on a copy of its curve pushed slightly off the surface into $S^3$, and such a copy is unknotted.\n\nThe theorem at work: $S^2 \\times S^1$ is $0$-surgery on the unknot, and the lens space $L(p, 1)$ is integral surgery with coefficient $\\pm p$ on the unknot, depending on the orientation convention.\nThe theorem implies that every closed orientable $3$-manifold bounds a compact orientable $4$-manifold, obtained from the $4$-ball by attaching $2$-handles along $L$.\n\n## Explanations\n\n### L1 human\n\nEvery closed orientable $3$-dimensional space can be built from the $3$-sphere by a finite number of drill-and-refill operations along circles.\nThe circles can even be taken unknotted, each refilled with a single twist.\nSo $3$-manifolds can be drawn as pictures of links with numbers attached.\n\n### L2 human\n\nThe theorem is the starting point of the description of $3$-manifolds by framed links, and of [[Kirby calculus|AVQ9WN1MDV]], which says when two framed links give the same manifold.\nQuantum invariants of $3$-manifolds, such as the Witten–Reshetikhin–Turaev invariants, are defined from link invariants through such surgery presentations.\nIts $4$-dimensional reading, that every $3$-manifold bounds a $4$-manifold built with $2$-handles, connects it with the vanishing of the oriented cobordism group in dimension $3$.\n",
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