Eta Invariants for Even Dimensional Manifolds

In previous work, we introduced eta invariants for even dimensional manifolds. It plays the same role as the eta invariant of Atiyah-Patodi-Singer, which is for odd dimensional manifolds. It is associated to $K^1$ representatives on even dimensional manifolds, and is defined on a finite cylinder, rather than on the manifold itself. Thus it is an interesting question to find an intrinsic spectral interpretation of this new invariant. Using adiabatic limit technique, we give such an intrinsic interpretation.