HIGHER ETA-INVARIANTS AND THE NOVIKOV-CONJECTURE ON MANIFOLDS WITH BOUNDARY

Let (N, g) be a closed Riemannian manifold of dimension 2m - 1 and Gam ma --> (N) over tilde --> N be a Galois covering of N. We assume that Gamma is of polynomial growth and that Delta((N) over tilde) is L-2-in vertible in degree m. By employing spectral sections which are symmetr ic with respect to the star-Hodge operator, we define the higher era i nvariant associated to 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 manif old with boundary (partial derivative M = N) then, under the above inv ertibility assumption on Delta(partial derivative (M) over tilde), we define a canonical Atiyah-Patodi-Singer signature-index class, in K-0 (C-r (Gamma)). Employing the higher APS index theory developed in [4] we express the Chern character of this index class in terms of a loca l integral and of the higher eta invariant defined above. We apply the se results to the problem of the existence and homotopy invariance of higher signatures on manifolds with boundary. (C) Academie des Science s/Elsevier, Paris.