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

  • Kevin Hutchinson

  • Liqun Tao

Abstract

Let FF be a field of characteristic zero and let ft,nf_{t,n} be the stabilization homomorphism from the nnth integral homology of SLt(F)\mathrm{SL}_t(F) to the nnth integral homology of SLt+1(F)\mathrm{SL}_{t+1}(F). We prove the following results: For all n,ft,nn, f_{t,n} is an isomorphism if tn+1t\geq n+1 and is surjective for t=nt=n, confirming a conjecture of C-H. Sah. fn,nf_{n,n} is an isomorphism when nn is odd and when nn is even the kernel is isomorphic to the (n+1)(n+1)st power of the fundamental ideal of the Witt Ring of FF. When nn is even the cokernel of fn1,nf_{n-1,n} is isomorphic to the nnth Milnor-Witt KK-theory group of FF. When nn is odd, the cokernel of fn1,nf_{n-1,n} is isomorphic to the square of the nnth Milnor KK-group of FF.