Dependence on the spin structure of the eta and Rokhlin invariants

We study the dependence of the eta invariant ηD on the spin structure, where D is a twisted Dirac operator on a (4k+3)-dimensional spin manifold. The difference between the eta invariants for two spin structures related by a cohomology class which is the reduction of a H1(M,Z)-class is shown to be a half integer. As an application of the technique of proof the generalized Rokhlin invariant is shown to be equal modulo 8 for two spin structures related in this way.