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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?

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up vote 15 down vote accepted

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.

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thanks alot for your clear answer! – Gian Jan 8 '12 at 10:52
Actually the relative eta invariant (aka the "rho"-invariant) is not always a homotopy invariant. It fails for example in the case of Lens spaces. Homotopy invariance for rho-invariants has been shown by N.Keswani for manifolds whose fundamental grous are torsion-free and satisfy the maximal version of the Baum-Connes conjecture. See this paper: The eta-invariant is nevertheless a diffeomorphism and a spectral invariant. – Indrava Roy Mar 5 '12 at 11:19
Sorry, you're quite right. I did not mean to make such an absolute statement, and I edited the answer accordingly. – Paul Siegel Mar 5 '12 at 14:43

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

I have computed the eta invariants for the $spin^c$ Dirac operators on Seifert $3$-manifolds.

See this paper for the special case of circle bundles. Here I describe in some detail how one goes about computing eta invariants (never easy) and I included some references about computations of the eta invariant that arises in the APS problem for the signature operator. For the more general case of Seifert manifolds see this paper.

The lens spaces mentioned by Paul Siegel are special cases of Seifert manifolds.

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