Kevin Hutchinson, Tao Liqun: Homology stability for the special linear group of a field and Milnor-Witt K-theory

Submission: 2008, Nov 18

Let F be a field of characteristic zero and let f(t,n) be the stabilization homomorphism from the n-th integral homology of SL(t,F) to the n-th homology of SL(t+1,F). We prove the following results: For all n, f(t,n) is an isomorphism if t is at least n+1, and is surjective for t=n, confirming a conjecture of C-H. Sah. Furthermore if n is odd, then f(n,n) is an isomorphism. If n is even, then the cokernel of f(n-1,n} is naturally isomorphic to the n-th Milnor-Witt $K$-group of F, MWK(n,F). This answers a question of Jean Barge and Fabien Morel. If n is odd there is a natural short exact sequence 0 -> Coker(f(n-1,n)) -> MWK(n,F) -> Ker(f(n-1,n-1)) -> 0.

2000 Mathematics Subject Classification: 19G99, 20G10

Keywords and Phrases: K-theory, Special linear group, Group Homology

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