ON THE THIRD HOMOLOGY OF SL2 AND WEAK HOMOTOPY INVARIANCE

The goal of the paper is to achieve - in the special case of the linear group SL2 - some understanding of the relation between group homology and its A(1)-invariant replacement. We discuss some of the general properties of the 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)-invariant version. In other words, weak homotopy invariance fails for SL2 over many families of non-algebraically closed fields.