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  "body": "\n## Summary\n\nA handle decomposition builds a manifold from a ball, or from a given boundary, by attaching handles $D^k \\times D^{n-k}$ one after another.\nEach handle is glued along part of its boundary.\nIt is the smooth analogue of a CW structure: a $k$-handle plays the role of a $k$-cell, thickened to full dimension.\nEvery compact smooth manifold has a handle decomposition, obtained from a [[Morse function|KPQ96640ZJ]], with one $k$-handle for each critical point of index $k$.\nHandle decompositions are the main tool in the proof of the h-cobordism theorem and in surgery theory.\nIn dimensions $3$ and $4$ they underlie Kirby calculus.\n\n## Explanations\n\n### L1 human\n\nBuild a surface out of clay: start with a disc, glue on flat strips along their two ends, and cap off with more discs.\nThe discs at the start are $0$-handles, the strips are $1$-handles, and the final caps are $2$-handles.\nA torus can be built from one disc, two strips and one cap.\nIn higher dimensions the pieces are thickened discs of the same dimension as the manifold, glued along part of their boundary.\n",
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