Split Milnor-Witt Motives and its Applications to Fiber Bundles

TL;DR

This paper investigates the structure of Milnor-Witt motives, providing a decomposition that relates motivic and Witt cohomology, and applies this to compute cohomology of Grassmannian and flag bundles, revealing 2-torsion properties.

Contribution

It introduces a splitting of Milnor-Witt motives into simpler components and applies this to derive cohomological properties of fiber bundles.

Findings

Milnor-Witt motives can be decomposed into direct sums involving motivic and Witt cohomology.

The paper defines motivic Bockstein cohomology and relates it to Witt cohomology.

The cohomology of real complete flags contains only 2-torsion elements.

Abstract

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 could 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 subgroup 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.