Counter-examples to a conjecture of Karpenko via truncated Brown-Peterson cohomology

TL;DR

This paper introduces a novel method connecting truncated Brown-Peterson cohomology with connective K-theory to construct counter-examples, disproving Karpenko's conjecture for new groups including the smallest known spinor group where it fails.

Contribution

The authors develop a new approach using cohomology theories to find counter-examples to Karpenko's conjecture, extending known failures to additional groups.

Findings

Disproved the conjecture for Spin_{15}

Established a new method for constructing counter-examples

Extended the list of groups where the conjecture fails

Abstract

Let $G$ be a split semisimple linear algebraic group and let $X$ denote the generically twisted variety of Borel subgroups in $G$. Nikita Karpenko conjectured that the map from the Chow ring of $X$ to the associated graded ring of the topological filtration on the Grothendieck ring of $X$ is an isomorphism. After having been verified for many $G$, the conjecture was disproved by Nobuaki Yagita for some spinor groups. Later, other counter-examples were constructed by Baek-Karpenko and Baek-Devyatov. We present a new method for constructing counter-examples that is based on the connection of the truncated Brown-Peterson cohomology with the connective K-theory. Using this method, we disprove the conjecture for new groups, including $\mathrm{Spin}_{15}$, which is now the smallest known spinor group for which the conjecture fails.