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On the third homology of $ SL_2$ and weak homotopy invariance


Authors: Kevin Hutchinson and Matthias Wendt
Journal: Trans. Amer. Math. Soc. 367 (2015), 7481-7513
MSC (2010): Primary 20G10; Secondary 14F42
DOI: https://doi.org/10.1090/S0002-9947-2014-06495-7
Published electronically: November 12, 2014
MathSciNet review: 3378837
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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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Additional Information

Kevin Hutchinson
Affiliation: School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland
Email: kevin.hutchinson@ucd.ie

Matthias Wendt
Affiliation: Fakultät Mathematik, Universität Duisburg-Essen, Thea-Leymann-Strasse 9, 45127, Essen, Germany
Email: matthias.wendt@uni-due.de

DOI: https://doi.org/10.1090/S0002-9947-2014-06495-7
Keywords: Weak homotopy invariance, group homology
Received by editor(s): October 18, 2013
Received by editor(s) in revised form: April 25, 2014
Published electronically: November 12, 2014
Article copyright: © Copyright 2014 American Mathematical Society

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