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Several proofs have been published of the Mod Z gluing formula for theeta-invariant of a Dirac operator. However, so far the integer contribution tothe gluing formula for the eta-invariant is left obscure in the literature. Inthis article we present a gluing formula for the eta-invariant which expressesthe integer contribution as a triple index involving the boundary conditionsand the Calderon projectors of the two parts of the decomposition. The mainingredients of our presentation are the Scott-Wojciechowski theorem for thedeterminant of a Dirac operator on a manifold with boundary and the approach ofBruning-Lesch to the mod Z gluing formula.  Our presentation includes careful constructions of the Maslov index andtriple index in a symplectic Hilbert space. As a byproduct we give intuitivelyappealing proofs of two theorems of Nicolaescu on the spectral flow of Diracoperators.  As an application of our methods, we carry out a detailed analysis of theeta-invariant of the odd signature operator coupled to a flat connection usingadiabatic methods. This is used to extend the definition of theAtiyah-Patodi-Singer rho-invariant to manifolds with boundary. We derive a``non-additivity'' formula for the Atiyah-Patodi-Singer rho-invariant andrelate it to Wall's non-additivity formula for the signature ofeven-dimensional manifolds.

The [eta]-invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary