Orientability of Fredholm families and topological degree

We construct a degree theory for oriented Fredholm mappings of index zero between open subsets of Banach spaces and between Banach manifolds. Our approach is based on the orientation of Fredholm mappings: it does not use Fredholm structures on the domain and target spaces. We provide a computable formula for the change in degree through an admissible homotopy that is necessary for applications to global bifurcation. The notion of orientation enables us to establish rather precise relationships between our degree and many other degree theories for particular classes of Fredholm maps, including the Elworthy-Tromba degree, which have appeared in the literature in a seemingly unrelated manner