  We compute eta invariants of various Dirac type operators on circle bundles
over Riemann surfaces via two approaches: an adiabatic approach based on the
results of Bismut-Cheeger-Dai and a direct elementary one. These results,
coupled with some delicate spectral flow computations are then used to
determine the virtual dimensions of Seiberg-Witten finite energy moduli spaces
on any 4-manifold bounding unions of circle bundles. This belated paper should
be regarded as the analytical backbone of dg-ga/9711006. There, we indicated
only what changes are needed to extend the methods of the present paper to
Seifert fibrations and we focused only to topological and number theoretic
aspects related to Froyshov invariants
