On third homology of SL_2 and weak homotopy invariance

TL;DR

This paper investigates the relationship between the third homology of SL_2 and its A^1-invariant version, revealing significant differences and failures of weak homotopy invariance over many fields.

Contribution

It provides new insights into the failure of weak homotopy invariance for SL_2 by analyzing group homology and A^1-invariant properties, using refined Bloch groups and stabilization sequences.

Findings

Large differences between group homology and A^1-invariant homology for SL_2.

Weak homotopy invariance fails over many non-algebraically closed fields.

A^1-invariant group homology exhibits complex stabilization and module structures.

Abstract

The goal of the paper is to achieve - in the special case of the linear group SL_2 - some understanding of the relation between group homology and its A^1-invariant replacement. We discuss some of the general properties of 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 to deduce that in general there is a rather large difference between group homology and its A^1-invariant version. In other words, weak homotopy invariance fails for SL_2 over many families of non-algebraically closed fields.