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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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