Integral Motivic Cohomology of $BSO_{4}$

TL;DR

This paper computes the integral motivic cohomology of the classifying space BSO_4, advancing the understanding of cohomological invariants in algebraic geometry and setting the stage for future related computations.

Contribution

It provides the first computation of the motivic cohomology of BSO_4 with integral coefficients, extending previous work on related classifying spaces.

Findings

Computed motivic cohomology of BSO_4 with integral coefficients.

Established methods for future computations of classifying spaces like BG_2.

Abstract

Motivic cohomology is powerful tool in algebraic geometry with associated realization maps giving important information about the relations between cohomological invariants of schemes and their classifying spaces. The problem of computing general cohomological invariants of these classifying spaces is ongoing. Most relevant to this paper is (1) Totaro's construction of the Chow ring of a classifying space in general and his use of this to study symmetric groups in arXiv:math/9802097, (2) Guillot's similar examination for the Lie groups $G_{2}$ and $Spin(7)$ in arXiv:math/0508122, (3) Field's computation of the Chow ring of $BSO(2n,\mathbb{C})$ in arXiv:math/0411424, and (4) Yagita's work on the $\mathbb{Z}_{2}$-motivic cohomology of $BSO_{4}$ and $BG_{2}$ in [Yag10]. The work presented in this paper covers the computation of the motivic cohomology of $BSO_{4}$ with integral coefficients. The primary approach draws on methods laid out by Guillot and Yagita (arXiv:math/0508122, [Yag10]). These results lay the groundwork for future work, most immediately the analogous computation for $BG_{2}$ ([Por21]).