Turning vector bundles

We define a turning of a rank-$2k$ vector bundle $E\to B$ to be a homotopy of bundle automorphisms ${\psi }_t$ from ${\mathbb{I}\mathbb{d}}_E$, the identity of $E$, to $-{\mathbb{I}\mathbb{d}}_E$, minus the identity, and call a pair $(E,{\psi }_t)$ a turned bundle. We investigate when vector bundles admit turnings and develop the theory of turnings and their obstructions. In particular, we determine which rank-$2k$ bundles over the $2k$-sphere are turnable. If a bundle is turnable, then it is orientable. In the other direction, complex bundles are turned bundles and for bundles over finite $CW$-complexes with rank in the stable range, Bott’s proof of his periodicity theorem shows that a turning of $E$ defines a homotopy class of complex structure on $E$. On the other hand, we give examples of rank-$2k$ bundles over $2k$-dimensional spaces, including the tangent bundles of some $2k$-manifolds, which are turnable but do not admit a complex structure. Hence turned bundles can be viewed as a generalisation of complex bundles. We also generalise the definition of turning to other settings, including other paths of automorphisms, and we relate the generalised turnability of vector bundles to the topology of their gauge groups and the computation of certain Samelson products.