  Let M be a riemannian manifold. The existence of a spin structure on M,
enables to study the topology of M. The obstruction to the existence of the
spin structure is given by the second Stiefel-Whitney class. This class is the
classifying cocycle of a gerbe. One may expect that the study of this gerbe may
have topological applications, for example, one may try to generalize the
spinors Lichnerowicz theorem in this setting. On this purpose, we must first
prove an Atiyah-Singer theorem for gerbes which is the main goal of this paper.
