Kreck-Stolz invariants for 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 $H^3(M; \mathbb Q) = 0$ 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 a 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 $\mathbb H P^k$. The $t$-invariant for $S^{4k-1}$ is closely related to the Adams $e$-invariant on the $(4k-5)$-stem.