Smale's degree for proper Fredholm maps

Remarks

The definition makes sense because regular values exist and the parity does not depend on $y$. Regular values form a residual subset of $N$ by the Sard–Smale theorem. At 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. The 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.

Example: $f \colon \R \to \R$, $f(x) = x^3$, is proper, and every smooth map between manifolds of the same finite dimension is Fredholm of index $0$. The value $y = 1$ is regular with the single preimage $x = 1$, so $\deg_2 f = 1$.

Non-example: $g \colon \R \to \R$, $g(x) = e^x$, is Fredholm of index $0$ but not proper. The 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.

In 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. For compact perturbations of the identity on a Banach space, the Leray–Schauder degree gives an integer-valued theory.