Degree theory for C1-Fredholm mappings of index 0

We develop a degree theory for C 1 Fredholm mappings of index 0 between Banach spaces and Banach manifolds. As in earlier work devoted to the C 2 case, our approach is based upon the concept of parity of a curve of linear Fredholm operators of index 0. This avoids considerations about Fredholm structures involved in other approaches and leads to a theory as complete as that of Leray-Schauder in a much broader setting. In particular, the well-known possible sign change under homotopy is fully elucidated. The technical difficulty arising with C a versus C 2 Fredholm mappings of index 0 is notorious: with only C 1 smoothness, the Sarff-Smale theorem is no longer available to handle crucial issues involving homotopy. In this work, this difficulty is overcome by using a new approximation theorem for C a Fredholm mappings of arbitrary index instead of the Sard-Smale theorem when dealing with homotopies.