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  "body": "\n## Summary\n\nA smooth structure on a [[topological manifold|QQQ9B1DK4P]] is a maximal atlas of charts whose transition maps are smooth.\nIt is the extra data that turns a topological manifold into a [[smooth manifold|F45VQ47GPV]], so that one can differentiate functions and maps on it.\nA topological manifold may have no smooth structure, exactly one up to diffeomorphism, or many.\nThe study of these possibilities is a central theme of differential topology.\n\n## Explanations\n\n### L1 human\n\nA topological manifold only tells you which points are near which.\nA smooth structure also tells you which functions on it count as smooth, by fixing a family of coordinate systems that fit together without kinks.\nFor example, the coordinate $t$ on the line $\\R$ gives the usual smooth structure.\nThe coordinate $t^3$ gives a different atlas, because $t \\mapsto t^{1/3}$ is not smooth at $0$, but the map $t \\mapsto t^{1/3}$ is a diffeomorphism between the two structures.\nSome spaces are more surprising: the $7$-sphere carries smooth structures that are not diffeomorphic to the standard one.\n",
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