Homology Stability for the Special Linear Group of a Field and Milnor-Witt $K$-theory

Abstract

Let $F$ be a field of characteristic zero and let $f_{t,n}$ be the stabilization homomorphism from the $n$th integral homology of $\mathrm{SL}_t(F)$ to the $n$th integral homology of $\mathrm{SL}_{t+1}(F)$. We prove the following results: For all $n, f_{t,n}$ is an isomorphism if $t\geq n+1$ and is surjective for $t=n$, confirming a conjecture of C-H. Sah. $f_{n,n}$ is an isomorphism when $n$ is odd and when $n$ is even the kernel is isomorphic to the $(n+1)$st power of the fundamental ideal of the Witt Ring of $F$. When $n$ is even the cokernel of $f_{n-1,n}$ is isomorphic to the $n$th Milnor-Witt $K$-theory group of $F$. When $n$ is odd, the cokernel of $f_{n-1,n}$ is isomorphic to the square of the $n$th Milnor $K$-group of $F$.