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  "body": "\n## Definition\n\nLet $M$ and $N$ be connected $C^2$ [[Banach manifolds|9K15AMGBV9]], with $M$ separable.\nLet $f \\colon M \\to N$ be a $C^2$ [[Fredholm map|GFXQ6HEM65]] of [[Fredholm index|2V9Q4KHY1T]] $0$ that is [[proper|15XEBGNGHC]].\nA point $y \\in N$ is a *regular value* of $f$ if $df_x$ is surjective for every $x \\in f^{-1}(y)$.\nFor a regular value $y$, the fibre $f^{-1}(y)$ is finite.\nThe *Smale degree* of $f$ is\n$$\\deg_2 f = \\# f^{-1}(y) \\bmod 2 \\in \\Z/2,$$\nfor any regular value $y$ of $f$.\n\n## Remarks\n\nThe definition makes sense because regular values exist and the parity does not depend on $y$.\nRegular values form a residual subset of $N$ by the [[Sard–Smale theorem|Z0NPFYDVP2]].\nAt a point of $f^{-1}(y)$ the derivative is a surjective Fredholm operator of index $0$, hence an isomorphism, so $f^{-1}(y)$ is discrete; properness makes it compact, hence finite.\nThe degree is also unchanged under a $C^2$ homotopy $H \\colon M \\times [0, 1] \\to N$ that is proper and is Fredholm of index $0$ at each time.\n\nExample: $f \\colon \\R \\to \\R$, $f(x) = x^3$, is proper, and every smooth map between manifolds of the same finite dimension is Fredholm of index $0$.\nThe value $y = 1$ is regular with the single preimage $x = 1$, so $\\deg_2 f = 1$.\n\nNon-example: $g \\colon \\R \\to \\R$, $g(x) = e^x$, is Fredholm of index $0$ but not proper.\nThe regular value $1$ has one preimage and the regular value $-1$ has none, so the parity depends on the point, and no degree is defined.\n\nIn finite dimensions, for compact connected manifolds of the same dimension, $\\deg_2 f$ is the mod $2$ reduction of the [[degree of a continuous map|TMFG2P912R]].\nFor compact perturbations of the identity on a Banach space, the [[Leray–Schauder degree|3VJC5Q9MYA]] gives an integer-valued theory.\n",
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