A higher Atiyah-Patodi-Singer index theorem for the signature operator on Galois coverings

Let (N, g) be a closed Riemannian manifold of dimension 2m - 1 and let Gamma --> (N) over tilde --> N be a Galois covering of N. We assume that Gamma is of polynomial growth with respect to a word metric and that Delta((N) over tilde) is L-2-invertible in degree m. By employing spectral sections with a symmetry property with respect to the star-Hodge operator, we define the higher eta invariant associated with the signature operator on (N) over tilde, thus extending previous work of Lott. If pi(1)(M) --> (M) over tilde --> M is the universal cover of a compact orientable even-dimensional manifold with boundary (partial derivative M = N) then, under the above invertibility assumption on Delta(partial derivative (M) over tilde) , and always employing symmetric spectral sections, we define a canonical Atiyah-Patodi-Singer index class, in K-0(C*(r) (Gamma)), for the signature operator of (M) over tilde. Using the higher APS index theory developed in [6], we express the Chern character of this index class in terms of a local integral and of the higher eta invariant defined above, thus establishing a higher APS index theorem for the signature operator on Galois coverings. We expect the notion of a symmetric spectral section for the signature operator to have wider implications in higher index theory for signatures operators.