Homology Stability for the Special Linear Group of a Field and Milnor-Witt K-theory Dedicated to Andrei Suslin

Let F be a field of characteristic zero and let ft,n be the stabilization homomorphism from the nth integral homology of SLt(F ) to the nth integral homology of SLt+1(F ). 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 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 the nth Milnor-Witt K-theory group of F . When n is odd, the cokernel of fn−1,n is isomorphic to the square of the nth Milnor K-group of F . 2010 Mathematics Subject Classification: 19G99, 20G10