The η–invariant, Maslov index, and spectral flow for Dirac–type operators on manifolds with boundary

Several proofs have been published of the modZ gluing formula for the η–invariant of a Dirac operator. However, so far the integer contribution to the gluing formula for the η–invariant is left obscure in the literature. In this article we present a gluing formula for the η–invariant which expresses the integer contribution as a triple index involving the boundary conditions and the Calderón projectors of the two parts of the decomposition. The main ingredients of our presentation are the Scott– Wojciechowski theorem for the determinant of a Dirac operator on a manifold with boundary and the approach of Brüning–Lesch to the modZ gluing formula. Our presentation includes careful constructions of the Maslov index and triple index in a symplectic Hilbert space. As a byproduct we give intuitively appealing proofs of two theorems of Nicolaescu on the spectral flow of Dirac operators. As an application of our methods, we carry out a detailed analysis of the η– invariant of the odd signature operator coupled to a flat connection using adiabatic methods. This is used to extend the definition of the Atiyah–Patodi–Singer ρ–invariant to manifolds with boundary. We derive a “non–additivity” formula for the Atiyah– Patodi–Singer ρ–invariant and relate it to Wall’s non-additivity formula for the signature of even–dimensional manifolds.