On third homology of SL₂ and weak homotopy invariance

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.