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  "body": "\n## Summary\n\nSmale's degree is a mod $2$ degree for proper [[Fredholm maps|GFXQ6HEM65]] of index $0$ between connected Banach manifolds.\nIt counts the preimages of a regular value modulo $2$.\nIt extends the mod $2$ [[degree of a continuous map|TMFG2P912R]] between compact manifolds of the same dimension to infinite dimensions.\nIt rests on the [[Sard–Smale theorem|Z0NPFYDVP2]], which supplies regular values.\n\n## Explanations\n\n### L1 human\n\nFor a map between two spaces of the same finite dimension, one can pick a typical point of the target and count how many points land on it.\nModulo $2$, this count does not depend on the point chosen, as long as the map is proper.\nSmale noticed that the same count works in infinite dimensions when the derivative of the map is a [[Fredholm operator|8HWGKWD6ZM]] of index $0$.\nSuch an operator behaves like a square matrix up to finite-dimensional errors.\nFor example, $x \\mapsto x^3$ on $\\R$ has one preimage over every point, so its degree is $1$.\nThe map $x \\mapsto x^2$ has two preimages over $1$ and none over $-1$, so its degree is $0$.\n",
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