A refinement of a conjecture of Quillen

TL;DR

This paper advances the understanding of the Quillen conjecture by providing new cohomological results for SL_2-groups, refining the conjecture based on partial positive results and known counterexamples.

Contribution

It introduces a refined version of the Quillen conjecture by combining new cohomology results with existing partial results and counterexamples.

Findings

Partial positive results for the Quillen conjecture in rank one.

New cohomology computations for SL_2-groups above the virtual cohomological dimension.

Formulation of a refined conjecture based on combined evidence.

Abstract

We present some new results on the cohomology of a large scope of SL\_2-groups in degrees above the virtual cohomological dimension; yielding some partial positive results for the Quillen conjecture in rank one. We combine these results with the known partial positive results and the known types of counterexamples to the Quillen conjecture, in order to formulate a refined variant of the conjecture.