Revisiting pinors and orientability
Let \(\widetilde X\) be the orientable double cover of a non-orientable manifold \(X\) equipped with the orientation-reversing involution \(\tau\) such that \(\widetilde X/\tau\simeq X\). The authors study the relations between pin structures on \(X\) and \(\tau\)-invariant pin structures on \(\widetilde X\). They show that there is not a simple bijection, but that the natural map induced by pull-back is neither injective nor surjective. By imposing additional conditions the authors recover the full correspondence. Moreover, the authors consider examples of surfaces, with detailed computations for the real projective plane, the Klein bottle and the Moebius strip.
scientific article; zbMATH DE number 6079013