Real line bundles on spheres

Abstract:

In a recent paper the author proved a classification theorem for Atiyah-real vector bundles on spaces with free involutions. This result is now applied to the group of Atiyah-real line (i.e., one-dimensional) bundles on spheres, denoted ${L_R}({S^n})$. It is proved that such bundles are classified by maps into a complex quadric $Q{C^n}$. Using this classification it is proved that ${L_R}({S^1}) = 0$ and that for $n \geqq 3$ the groups are all isomorphic.