Kreck-Stolz invariants for quaternionic line bundles

Abstract

We generalise the Kreck-Stolz invariants s 2 s_2 and s 3 s_3 by defining a new invariant, the t t -invariant, for quaternionic line bundles  E E over closed spin-manifolds M M of dimension 4 k − 1 4k-1 with  H 3 ( M ; Q ) = 0 H^3(M; \mathbb Q) = 0 such that  c 2 ( E ) ∈ H 4 ( M ) c_2(E)\in H^4(M) is torsion. The t t -invariant classifies closed smooth oriented 2 2 -connected rational homology 7 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 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 H P k \mathbb H P^k . The t t -invariant for S 4 k − 1 S^{4k-1} is closely related to the Adams e e -invariant on the ( 4 k − 5 ) (4k-5) -stem.