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

TL;DR

This paper generalizes the adiabatic limit formula for eta-invariants to Seifert fibrations, incorporating Dedekind sums from singular fibers, and applies it to classify certain 7-manifolds including a new positively curved example.

Contribution

It extends the adiabatic limit formula to Seifert fibrations with singular fibers, introducing Dedekind sums, and applies this to compute invariants and classify specific 7-manifolds.

Findings

Derived a new formula for eta-invariants with singular fibers

Computed invariants for cohomogeneity one manifolds

Determined the diffeomorphism type of a new positively curved 7-manifold

Abstract

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.