The third homology of the special linear group of a field

Abstract

We prove that for any infinite field homology stability for the third integral homology of the special linear groups $SL(n,F)$ begins at $n=3$. When $n=2$ the cokernel of the map from the third homology of $SL(2,F)$ to the third homology of $SL(3,F)$ is naturally isomorphic to the square of Milnor $K_3$. We discuss applications to the indecomposable $K_3$ of the field and to Milnor-Witt K-theory.