  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.
