Odd rank vector bundles in eta-periodic motivic homotopy theory

TL;DR

This paper explores the behavior of odd rank vector bundles in eta-periodic motivic homotopy theory, revealing their near-vanishing section properties and computing classifying spaces of algebraic groups.

Contribution

It introduces new insights into odd rank vector bundles and computes classifying spaces within eta-periodic motivic homotopy theory, advancing understanding of motivic algebraic topology.

Findings

Odd rank vector bundles behave as if they have a nowhere vanishing section in eta-periodic setting.

Computed classifying spaces of diagonalisable groups in eta-periodic motivic homotopy category.

Discussed implications for SLc-orientations and etale classifying spaces.

Abstract

We observe that, in the eta-periodic motivic stable homotopy category, odd rank vector bundles behave to some extent as if they had a nowhere vanishing section. We discuss some consequences concerning SLc-orientations of motivic ring spectra, and the etale classifying spaces of certain algebraic groups. In particular, we compute the classifying spaces of diagonalisable groups in the eta-periodic motivic stable homotopy category.