Singularities of Fredholm maps with one-dimensional kernels, I: A complete classification

Abstract

We study the simplest singular points of Fredholm maps of index zero between Banach spaces, i.e. when the kernel of the Frechet derivative of the map has dimension one. Even in this relatively simple case we have a rich variety of singularities which are completely classified under the natural geometric assumption of transversality of the map. In fact we have, locally, a suitable stratification of the singular points that allows us to identify three kinds of singularities: a) the ordinary or k-singularities (the infinite-dimensional analogues of the well-known Morin singularities) and two new types, b) the maximal-transverse singularities; c) the infinite-transverse singularities (the latter ones can only occur in infinite dimensions).