Odd rank vector bundles in eta-periodic motivic homotopy theory

Abstract: We observe that, in the eta-periodic motivic stable homotopy category, odd rank vector bundles behave to some extent as if they had a nowhere vanishing section. We discuss some consequences concerning SLc-orientations of motivic ring spectra, and the etale classifying spaces of certain algebraic groups. In particular, we compute the classifying spaces of diagonalisable groups in the eta-periodic motivic stable homotopy category.