A signature theorem for disk bundles and the eta invariant

Abstract

We apply the Signature Theorem of Atiyah, Patodi and Singer [2] to disk bundles over compact Riemannian manifolds. We show that the signature of a disk bundle can be expressed as the integral of a characteristic class over the manifold and a limiting eta invariant of the bounding sphere bundle. The characteristic class can be explicitly described and is modelled after the Hirzebruch L-class. The limiting process is essential and we show by a counter example that in the absence of the limit, the eta invariant is not a topological invariant. If the base manifold is Hodge, we obtain an expression for the limiting eta invariant in terms of our characteristic class and the bigraded Betti numbers of the base.