Dependence on the spin structure of the eta and rokhlin invariants

We study the dependence of the eta invariant $\eta_D$ 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 cohomolgy class which is the reduction of a $H^1(M,\Za)$-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.