Kreck-Stolz invariants for quaternionic line bundles

TL;DR

This paper introduces a new invariant called the t-invariant for quaternionic line bundles over certain spin-manifolds, which classifies specific 7-dimensional manifolds and relates to exotic smooth structures and the Adams e-invariant.

Contribution

It generalizes Kreck-Stolz invariants by defining the t-invariant, providing a classification tool for rational homology spheres and new insights into quaternionic line bundles.

Findings

The t-invariant classifies 2-connected rational homology 7-spheres up to almost-diffeomorphism.

It detects exotic homeomorphisms between such manifolds.

Provides a new proof of a theorem on second Chern classes of quaternionic line bundles.

Abstract

We generalise the Kreck-Stolz invariants s_2 and s_3 by defining a new invariant, the t-invariant, for quaternionic line bundles E over closed spin-manifolds M of dimension 4k-1 with H^3(M; \Q) = 0 such that c_2(E)\in H^4(M) is torsion. The t-invariant classifies closed smooth oriented 2-connected rational homology 7-spheres up to almost-diffeomorphism, that is, diffeomorphism up to connected sum with an exotic sphere. It also detects exotic homeomorphisms between such manifolds. The t-invariant also gives information about quaternionic line bundles over a fixed manifold and we use it to give a new proof of a theorem of Feder and Gitler about the values of the second Chern classes of quaternionic line bundles over HP^k. The t-invariant for S^{4k-1} is closely related to the Adams e-invariant on the (4k-5)-stem.