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On the third homology of $$\mathrm{SL}_2$$ and weak homotopy invariance. (English) Zbl 1330.20069
Summary: The goal of the paper is to achieve – in the special case of the linear group $$\text{SL}_2$$ – 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 $$\text{SL}_2$$ over many families of non-algebraically closed fields.

##### MSC:
 20G10 Cohomology theory for linear algebraic groups 14F42 Motivic cohomology; motivic homotopy theory 19D55 $$K$$-theory and homology; cyclic homology and cohomology 19D25 Karoubi-Villamayor-Gersten $$K$$-theory 19C09 Central extensions and Schur multipliers 55N35 Other homology theories in algebraic topology
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