Higher eta invariants and the Novikov conjecture on manifolds with boundary

Let ( N,g ) be a closed Riemannian manifold of dimension 2 m -1 and Γ → N ˜ → N be a Galois covering of N . We assume that Γ is of polynomial growth and that Δ N ˜ is L 2 -invertible in degree m . By employing spectral sections which are symmetric with respect to the *-Hodge operator, we define the higher eta invariant associated to the signature operator on N ˜ , thus extending previous work of Lott. If π 1 ( M ) → M ˜ → M is the universal cover of a compact orientable even-dimensional manifold with boundary ( ∂M = N ) then, under the above invertibility assumption on Δ ∂ M ˜ , we define a canonical Atiyah-Patodi-Singer signature-index class, in K 0 ( C r * (Γ)). Employing the higher APS index theory developed in [4] we express the Chern character of this index class in terms of a local integral and of the higher eta invariant defined above. We apply these results to the problem of the existence and homotopy invariance of higher signatures on manifolds with boundary.