On the third homology of 𝑆𝐿₂ and weak homotopy invariance

Hutchinson, Kevin; Wendt, Matthias

doi:10.1090/S0002-9947-2014-06495-7

On the third homology of $SL_2$ and weak homotopy invariance
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by Kevin Hutchinson and Matthias Wendt PDF

Trans. Amer. Math. Soc. 367 (2015), 7481-7513 Request permission

Abstract:

The goal of the paper is to achieve - in the special case of the linear group $SL_2$ - some understanding of the relation between group homology and its $\mathbb {A}^1$-invariant replacement. We discuss some of the general properties of the $\mathbb {A}^1$-invariant group homology, such as stabilization sequences and Grothendieck-Witt module structures. Together with very precise knowledge about refined Bloch groups, these methods allow us to deduce that in general there is a rather large difference between group homology and its $\mathbb {A}^1$-invariant version. In other words, weak homotopy invariance fails for $SL_2$ over many families of non-algebraically closed fields.

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