On \(Sp(2)\) and \(Sp(2)\cdot Sp(1)\)-structures in 8-dimensional vector bundles
Given an oriented \(8\)-plane bundle \(\xi\) over a CW-complex, the authors associate to it a certain other vector bundle \(\zeta\). They come to the conclusion that the structure group \(SO(8)\) of \(\xi\) can be reduced (a) to \(Sp(2)\cdot Sp(1)\) if and only if \(\zeta\) contains an oriented \(3\)-plane bundle, and (b) to \(Sp(2)\) if and only if \(\zeta\) has \(3\) everywhere linearly independent sections. In their main result, for \(\xi\) an oriented \(8\)-plane bundle over a closed connected spin \(8\)-manifold, they present the condition of (b) expressed in terms of Stiefel-Whitney classes, Pontryagin classes, and the Euler class of \(\xi\).
scientific article; zbMATH DE number 1146190