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

TL;DR

This paper computes the Chow--Witt rings of classifying spaces for quadratically oriented bundles and explores their cobordism spectra, revealing their structure and relationships after inverting certain elements.

Contribution

It provides explicit calculations of Chow--Witt rings for quadratically oriented bundles and analyzes the associated cobordism spectrum, connecting it to known spectra after inverting eta.

Findings

Chow--Witt rings of ${ m BSL}_n^c$ are explicitly computed.

The quadratically-oriented cobordism spectrum ${ m MSL}^c$ is shown to be equivalent to ${ m MSL}$ after inverting eta.

The results deepen understanding of quadratic orientations in algebraic topology.

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 $\eta$.