Let (N, g) be a closed Riemannianmanifold of dimension 2m − 1 and let Γ → Ñ → N be a Galois covering of N. We assumethat Γ is of polynomial growth with respect to a word metric and that ΔÑ is L2-invertible in degree m. By employing spectral sections with asymmetry property with respect to the ⋆-Hodge operator, we define the higher eta invariant associatedwith the signature operator on Ñ, thus extending previous work of Lott. If π1(M)→ M~M~ →M is the universal cover of a compact orientable even-dimensionalmanifold with boundary (∂M = N)then, under the above invertibility assumption on Δ∂ M~M~ , andalways employing symmetric spectral sections, we define acanonical Atiyah–Patodi–Singer index class, in K0(C*r(Γ)), for the signature operator of M~M~ . Using the higherAPS index theory developed in [6], we express the Chern character ofthis index class in terms of a local integral and of the higher etainvariant defined above, thus establishing a higher APS index theoremfor the signature operator on Galois coverings. We expect the notion ofa symmetric spectral section for the signature operator to have widerimplications in higher index theory for signatures operators.

Atiyah–Patodi–Singer higher index theoryb-pseudodifferential calculusGalois coveringshigher eta invariantshigher signaturessignature operatorsymmetric spectral sections