ODD RANK VECTOR BUNDLES IN ETA-PERIODIC MOTIVIC HOMOTOPY THEORY

Abstract

AbstractWe 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$\operatorname {\mathrm {SL}}^c$-orientations of motivic ring spectra and the étale classifying spaces of certain algebraic groups. In particular, we compute the classifying spaces of diagonalisable groups in the eta-periodic motivic stable homotopy category.