Nonlinear Fredholm maps of index zero and their singularities

Let $F:X \to Y$ be a ${C^1}$ Fredholm map of index zero between two Banach spaces. Defining the singular set $B = \{ x\vert F'(x)$ is not surjective}, we study the local and global effect of B on the map F. In particular it is shown that if $b \in B$ is isolated in B, then, for $\dim X$ and $\dim Y \geqslant 3$, F is a local homeomorphism at b. We then show that if B consists of discrete points, F is a global homeomorphism of X onto Y. A nonlinear partial differential equation is included as an illustration.