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  "body": "\n## Definition\n\nLet $M$ be a smooth manifold of dimension $n$, and let $p, q \\ge 0$ be integers with $p + q = n$.\nLet $\\varphi \\colon S^p \\times D^q \\to M$ be a smooth embedding, where $S^p$ is the $p$-[[sphere|T62369RFH6]] and $D^q$ the closed $q$-disc.\nThe *surgery* on $M$ along $\\varphi$ is the following manifold:\n$$M' = \\bigl(M \\setminus \\varphi(S^p \\times \\operatorname{int} D^q)\\bigr) \\cup_{\\varphi} \\bigl(D^{p+1} \\times S^{q-1}\\bigr).$$\nHere the two pieces are glued along their common boundary $S^p \\times S^{q-1}$ by $\\varphi$, and corners are smoothed.\nThe number $p$ is the index of the surgery.\n*Surgery theory* is the study of manifolds through such operations.\nIts central problem is to turn a degree one normal map into a homotopy equivalence by a sequence of surgeries.\n\n## Remarks\n\nThe *trace* of the surgery is $W = (M \\times [0, 1]) \\cup_{\\varphi \\times \\set{1}} (D^{p+1} \\times D^q)$, a $(p+1)$-handle attached to $M \\times [0,1]$.\nIt is a [[cobordism|T8BVGPPSZP]] from $M$ to $M'$, and every compact cobordism is obtained by a finite sequence of such traces.\nExample: let $M = S^1 \\times S^1$ and $\\varphi$ an embedding of $S^1 \\times D^1$ as an annulus around the meridian circle.\nRemoving the open annulus leaves a cylinder $S^1 \\times [0,1]$, and gluing in two discs $D^2 \\times S^0$ caps it to $S^2$.\nSo surgery of index $1$ on a torus along a meridian gives the sphere.\nNon-example: let $C$ be the core circle of a Möbius band inside $\\R P^2$.\nEvery tubular neighbourhood of $C$ is a Möbius band and not an annulus, so $C$ has no neighbourhood of the form $S^1 \\times D^1$.\nSo there is no embedding $\\varphi$ as in the definition with core $C$, and no surgery along $C$ is defined.\n\n## Explanations\n\n### L1 human\n\nTo do surgery, find a sphere with a product neighbourhood inside a manifold and cut that neighbourhood out.\nThen fill the hole with the other natural piece that has the same boundary.\nSurgery along a non-separating circle in a surface removes a handle; surgery along two points adds one.\nThe operation changes the topology in a controlled way, and the trace records it as a cobordism.\n",
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