Vector bundles and cohomotopies of spin 5-manifolds

The purpose of this paper is two-fold: On the one side we would like to close a gap on the classification of vector bundles over $5$-manifolds. Therefore it will be necessary to study quaternionic line bundles over $5$-manifolds which are in $1-1$ correspondence to elements in the first cohomotopy group $\pi^4(M)=[M,S^4]$ of $M$. From previous results this group fits into a short exact sequence, which splits into $H^4(M;\mathbb Z)\oplus\mathbb Z_2$ if $M$ is spin. The second intent is to provide a bordism theoretic splitting map for this short exact sequence, which will lead to a $\mathbb Z_2$-invariant for quaternionic line bundles. This invariant is related to the generalized Kervaire semi-characteristic.