Motivic cohomology of the Nisnevich classifying space of even Clifford groups

TL;DR

This paper computes the mod 2 motivic cohomology ring of the Nisnevich classifying space of split even Clifford groups, highlighting differences from spin groups in the behavior of the second subtle Stiefel-Whitney class.

Contribution

It provides the first explicit computation of the motivic cohomology for even Clifford groups, revealing key differences from spin groups.

Findings

The motivic cohomology ring is similar to that of spin groups.

The second subtle Stiefel-Whitney class is non-trivial for even Clifford groups.

The results highlight fundamental differences in cohomological properties between even Clifford and spin groups.

Abstract

In this paper, we consider the split even Clifford group $\Gamma^+_n$ and compute the mod 2 motivic cohomology ring of its Nisnevich classifying space. The description we obtain is quite similar to the one provided for spin groups in [11]. The fundamental difference resides in the behaviour of the second subtle Stiefel-Whitney class that is non-trivial for even Clifford groups, while it vanished in the spin-case.