The third homology of the special linear group of a field

TL;DR

This paper establishes homology stability for the third integral homology of special linear groups over infinite fields starting at n=3, and explores related algebraic K-theory implications.

Contribution

It proves homology stability for $SL(n,F)$ at n=3 over infinite fields and characterizes the cokernel at n=2, linking to Milnor K-theory.

Findings

Homology stability begins at n=3 for infinite fields.

Cokernel at n=2 is isomorphic to the square of Milnor K_3.

Applications to indecomposable K_3 and Milnor-Witt K-theory.

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.