Homology Stabilty 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. Furthermore if n is odd, then fn,n is an isomorphism. If n ≥ 1 is even then the cokernel of fn−1,n is naturally isomorphic to the nth Milnor-Witt K-group, K n (F ). This answers a question of Jean Barge and Fabien Morel. If n ≥ 3 is odd there is a natural short exact sequence 0→ Coker(fn−1,n)→ K n (F )→ Ker(fn−1,n−1)→ 0.