The general question is: What is known about the eta invariants of fiber bundles?
The particular case I am interested in is the following. The fiber bundle is a bundle $S$ of even-dimensional round spheres of constant radius. Let $V$ be the vector bundle associated to $S$. There is a spin structure on $TV$, so we obtain a Dirac operator on $S$ and one on the base.
At least in the situation outlined above, my (maybe naive) hope that there exists a formula expressing the eta invariant of the Dirac operator on $S$ in terms of the eta invariant of the Dirac operator on the base and characteristic classes of the sphere bundle. If there are good reasons why this is too much to hope for, I'd like to hear about them.
The only references I have been able to find so far are papers by Donnelly, Tiwari and Bohn that concern exclusively the eta invariant of the signature Dirac operator.
