# Chow-Witt rings of classifying spaces for symplectic and special linear groups

## TL;DR

This paper computes the Chow-Witt rings of classifying spaces for symplectic and special linear groups, revealing their structure and orientation properties, and discusses implications for vector bundle splitting.

## Contribution

It provides explicit computations of Chow-Witt rings for these classifying spaces, highlighting their symplectic orientation and extending cohomology understanding.

## Key findings

- Chow-Witt rings are symplectically oriented cohomology theories.
- Explicit cohomology computations for symplectic and special linear groups.
- Insights into splitting of odd-rank vector bundles.

## Abstract

We compute the Chow-Witt rings of the classifying spaces for the symplectic and special linear groups. In the structural description we give, contributions from real and complex realization are clearly visible. In particular, the computation of cohomology with $\mathbf{I}^j$-coefficients is done closely along the lines of Brown's computation of integral cohomology for special orthogonal groups. The computations for the symplectic groups show that Chow-Witt groups are a symplectically oriented ring cohomology theory. Using our computations for special linear groups, we also discuss the question when an oriented vector bundle of odd rank splits off a trivial summand.