Fredholm map

Remarks

The Fredholm operators form an open set on which the index is locally constant, and $x \mapsto d f_x$ is continuous in local charts. So the index is locally constant in $x$, and on a connected $M$ it is a single integer.

Example: every $C^1$ map $f \colon \R^m \to \R^n$ is Fredholm of index $m - n$, since $\dim \ker A - \dim \operatorname{coker} A = m - n$ for every linear map $A \colon \R^m \to \R^n$ by rank-nullity.

Non-example: the inclusion $\iota \colon \ell^2 \to \ell^2 \times \ell^2$, $x \mapsto (x, 0)$, is smooth and its derivative is injective with closed image, but the cokernel is isomorphic to $\ell^2$, which is infinite-dimensional, so $\iota$ is not Fredholm.

A map of the form $f = L + K$ with $L$ a linear Fredholm operator and $K$ a $C^1$ map whose derivatives are compact operators is Fredholm with the index of $L$.