# Subtle characteristic classes for $Spin$-torsors

## TL;DR

This paper provides a complete description of the motivic cohomology of the classifying spaces of spin groups and G2, revealing simple relations among subtle Stiefel-Whitney classes and connecting algebraic and topological results.

## Contribution

It extends previous work to fully characterize motivic cohomology of Spin_n and G_2 classifying spaces, highlighting relations among subtle characteristic classes.

## Key findings

- Explicit motivic cohomology descriptions for Spin_n and G_2.
- Simple relations among subtle Stiefel-Whitney classes.
- Connection between motivic and topological cohomology results.

## Abstract

Extending [14], we obtain a complete description of the motivic cohomology with ${\mathbb Z}/2$-coefficients of the Nisnevich classifying space of the spin group $Spin_n$ associated to the standard split quadratic form. This provides us with very simple relations among subtle Stiefel-Whitney classes in the motivic cohomology of \v{C}ech simplicial schemes associated to quadratic forms from $I^3$, which are closely related to $Spin_n$-torsors over the point. These relations come from the action of the motivic Steenrod algebra on the second subtle Stiefel-Whitney class. Moreover, exploiting the relation between $Spin_7$ and $G_2$, we describe completely the motivic cohomology ring of the Nisnevich classifying space of $G_2$. The result in topology was obtained by Quillen in [13].