Abstract: The Chow-Witt group is a kind of cohomology theory of smooth varieties which encodes information of singular cohomology of both real and complex points together. It admits significant applications in classifying vector bundles and Hermitian K-theory.
Recently, B. Calmès, F. Déglise and J. Fasel defined a kind of motivic theory representing the Chow-Witt groups, namely the Milnor-Witt (abbr. MW) motives. It's rationally equivalent to the motivic stable homotopy category defined by F. Morel.
In this talk, we introduce the notion of split MW-motives. Varieties whose MW-motives split have only 2-torsions in cohomology, so we use Bockstein cohomology to compute the torsion free component. We compute the MW-motive of Grassmannians bundles and complete flags bundles, which turn out to fit the split pattern we desired.
