Nonlinear fredholm maps of index zero and their singularities

Let F: X → Y be a C¹ Fredholm map of index zero between two Banach spaces. Defining the singular set B = (x|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 ∊ B is isolated in B, then, for dim X and dim Y ≥ 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.