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  "body": "\n## Summary\n\nSurgery is an operation that cuts out a thickened sphere $S^p \\times D^q$ from a manifold and glues in $D^{p+1} \\times S^{q-1}$ along the common boundary.\nThe result is a new manifold, [[cobordant|T8BVGPPSZP]] to the old one.\nSurgery theory uses this operation to decide whether a space is homotopy equivalent to a manifold.\nIt also classifies the manifolds within a homotopy type.\nIn dimensions at least $5$ it reduces these questions to homotopy theory and to algebra in the surgery obstruction groups.\n\n## Explanations\n\n### L1 human\n\nSurgery is a cut-and-paste move on a manifold.\nIn a surface, cut out two small discs and glue in a tube between the two holes: the surface gains a handle.\nThis is surgery on a $0$-sphere, the pair of centres of the discs.\nRepeating such moves, one can kill unwanted loops and holes and steer a manifold towards a desired shape.\nSurgery theory is the systematic study of when this steering succeeds.\n",
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