The Chow--Witt rings of the classifying space of quadratically oriented bundles

TL;DR

The paper computes the Chow–Witt rings of the classifying space ${ m BSL}_n^c$ of quadratically oriented rank-$n$ bundles by a fiber-product framework that couples Chow groups with ${f I}^j$-cohomology, and it models ${ m BSL}_n^c$ as an ind-variety via metalinear Grassmannians. It provides a detailed description of the Witt-sheaf and ${f I}^j$-cohomology, including generators such as even Pontryagin classes $p_{2i}$, the Euler class $e_n$, and Bockstein elements $eta(ar c_J)$ and $eta_ heta(ar c_J)$, together with a precise multiplication structure obtained from a fiber-product setup. The paper then connects these algebraic invariants to the real and η-inverted motivic worlds, showing that ${ m MSL}^c$ is η-inverted equivalent to ${ m MSL}$, and establishing real realizations ${ m Re}_{bR}({ m MSL}^c)[1/2] o{ m MSO}[1/2]$ and ${ m Re}_{bR}({ m MSL})[1/2] o{ m MSO}[1/2]$. Altogether, the results bridge quadratic orientation in algebraic geometry with classical topological cobordism and clarify how orientation data influences Chow–Witt theory and motivic homotopy theory.

Abstract

In this paper we compute the Chow--Witt rings of the classifying space ${\rm BSL}_n^c$ of quadratically oriented vector bundles of rank $n$. We also discuss the corresponding quadratically-oriented cobordism spectrum ${\rm MSL}^c$ and show that it is equivalent to $\rm{MSL}$ after inverting $η$.

Theorems & Definitions (87)

• Theorem 1.1
• Proposition 1.2
• Corollary 1.3
• Example 2.4
• Remark 2.5
• Proposition 2.7
• Lemma 2.9
• Remark 2.10
• Proposition 2.13
• Lemma 2.14