On the third homology of 𝑆𝐿₂ and weak homotopy invariance

Abstract

The goal of the paper is to achieve - in the special case of the linear group S L 2 SL_2 - some understanding of the relation between group homology and its A 1 \mathbb {A}^1 -invariant replacement. We discuss some of the general properties of the A 1 \mathbb {A}^1 -invariant group homology, such as stabilization sequences and Grothendieck-Witt module structures. Together with very precise knowledge about refined Bloch groups, these methods allow us to deduce that in general there is a rather large difference between group homology and its A 1 \mathbb {A}^1 -invariant version. In other words, weak homotopy invariance fails for S L 2 SL_2 over many families of non-algebraically closed fields.