  We extend the adiabatic limit formula for eta-invariants by Bismut-Cheeger
and Dai to Seifert fibrations. Our formula contains a new contribution from the
singular fibres that takes the form of a generalised Dedekind sum.
  As an application, we compute the Eells-Kuiper and t-invariants of certain
cohomogeneity one manifolds that were studied by Dearricott, Grove, Verdiani,
Wilking, and Ziller. In particular, we determine the diffeomorphism type of a
new manifold of positive sectional curvature.
