Geodesic
Homology stability for the special linear group of a field and Milnor-Witt $K$-theory
Documenta mathematica, Andrei A. Suslin's Sixtieth Birthday (2010), pp. 267-315.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Let $F$ be a field of characteristic zero and let $f_{t,n}$ be the stabilization homomorphism from the $n$th integral homology of $\mathrm{SL}_t(F)$ to the $n$th integral homology of $\mathrm{SL}_{t+1}(F)$. We prove the following results: For all $n, f_{t,n}$ is an isomorphism if $t\geq n+1$ and is surjective for $t=n$, confirming a conjecture of C-H. Sah. $f_{n,n}$ 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 $f_{n-1,n}$ is isomorphic to the $n$th Milnor-Witt $K$-theory group of $F$. When $n$ is odd, the cokernel of $f_{n-1,n}$ is isomorphic to the square of the $n$th Milnor $K$-group of $F$.
Classification : 19G99, 20G10
Keywords: $K$-theory, special linear group, group homology 
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     author = {Hutchinson, Kevin and Tao, Liqun},
     title = {Homology stability for the special linear group of a field and {Milnor-Witt} $K$-theory},
     journal = {Documenta mathematica},
     pages = {267--315},
     publisher = {mathdoc},
     volume = {Andrei A. Suslin's Sixtieth Birthday},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2010__S4__a12/}
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Hutchinson, Kevin; Tao, Liqun. Homology stability for the special linear group of a field and Milnor-Witt $K$-theory. Documenta mathematica, Andrei A. Suslin's Sixtieth Birthday (2010), pp. 267-315. http://geodesic.mathdoc.fr/item/DOCMA_2010__S4__a12/