Split Milnor–Witt motives and its applications to fiber bundles

We study the Milnor–Witt motives which are a finite direct sum of $\mathbb{Z}(q)[p]$ and $\mathbb{Z}/\eta(q)[p]$. 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.

Keywords

MW-motives, Grassmannian bundles, complete flag bundles

2010 Mathematics Subject Classification

14F42, 14N15

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Received 27 January 2021

Published 21 October 2022