Split Milnor–Witt motives and its applications to fiber bundles

Abstract

. We study the Milnor-Witt motives which are a finite direct sum of Z ( q )[ p ] and Z /η ( q )[ p ]. We show that for MW-motives of this type, we can determine an MW-motivic cohomology class in terms of a motivic cohomology class and a Witt cohomology class. We define the motivic Bockstein cohomology and show that it corresponds to subgroups of Witt cohomology, if the MW-motive splits as above. As an application, we give the splitting formula of Milnor-Witt motives of Grassmannian bundles and complete flag bundles. This in particular shows that the integral cohomology of real complete flags has only 2-torsions.