Homology stability for the special linear group of a field and Milnor-Witt K-theory

Let F be a field of characteristic zero and let ft,n be the stabilization homomorphism Hn(SLt(F), Z) → Hn(SLt+1(F), Z). We prove the following results: For all n, ft,n is an isomorphism if t ≥ n + 1 and is surjective for t = n, confirming a conjecture of C-H. Sah. fn,n is an isomorphism when n is odd and when n is even the kernel is isomorphic to I n+1(F), the (n + 1)st power of the fundamental ideal of the Witt Ring of F. When n is even the cokernel of fn−1,n is isomorphic to KMW n (F), the nth Milnor-Witt K-theory group of F. When n is odd, the cokernel of fn−1,n is isomorphic to 2KM n (F), where KM n (F) is the nth Milnor K-group of F.