Adiabatic limit of Seifert fibrations, Dedekind sums, and the diffeomorphism type of certain 7-manifolds

Eta invariants arise as a boundary correction term in the Atiyah-Patodi-Singer index theorem for manifolds with boundary. They also form the basic ingredients for several differential-topological invariants. I will explain the adiabatic limit formula for eta invariants on generalised Seifert fibrations and use it to determine the diffeomorphism types of certain cohomogeneity one manifolds, some of which admit positive sectional curvature.