I am trying to understand the Atiyah-Patodi-Singer index theorem in the case of Dirac operators in four dimensions. I have three questions about the eta invariant: 


1) Is eta a topological invariant (or geometric invariant)? 


2) Which is its relation with the three dimensional Chern-Simons form?


3) In how many non-trivial cases the eta invariant is explicitly calculable?



Atiyah-Patodi-Singer Eta invariant and Chern-Simons form
1) The eta invariant itself depends on the metric, but the *relative* eta invariant is in many cases (see comments) a homotopy invariant. The relative eta invariant is defined to be the difference of the eta invariants associated to the Dirac operator twisted by two different flat Hermitian bundles (i.e. unitary representations of the fundamental group).


2) The relation between the eta invariant and Chern-Simons invariants is a little bit subtle, but it is explained in detail in section 4 of "Spectral Asymmetry and Riemannian Geometry II" by A-P-S.


3) Arguably the most important examples are lens spaces - this is how it was first realized that the defect in the signature theorem for manifolds with boundary is non-local, for example (if it were local it would be multiplicative for coverings). There is also an interesting paper called "Eta Invariants, Signature Defects of Cusps, and Values of L-Functions" by Atiyah, Donnelly, and Singer in which the eta invariant associated to the signature operator on a Hilbert modular variety with the cusps chopped off is calculated in terms of values of Shimazu L-functions. This was apparently one of the motivating examples for the theory of eta invariants, but I don't know what actual arithmetic significance it has.


