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 0→ Ñ → N be a Galois covering of N . We assume that 0 is of polynomial growth with respect to a word metric and that 1Ñ is L 2-invertible in degree m. By employing spectral sections with a symmetry property with respect to the ?-Hodge operator, we define the higher eta invariant associated with the signature operator on Ñ , thus extending previous work of Lott. If π1(M)→ M̃ → M is the universal cover of a compact orientable even-dimensional manifold with boundary (∂M = N) then, under the above invertibility assumption on 1∂M̃ , and always employing symmetric spectral sections, we define a canonical Atiyah–Patodi–Singer index class, in K0(C ∗ r (0)), for the signature operator of M̃. 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. Mathematics Subject Classifications (1991): 58G12, 19D55.