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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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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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
  • Received by editor(s): October 18, 2013
  • Received by editor(s) in revised form: April 25, 2014
  • Published electronically: November 12, 2014
  • © Copyright 2014 American Mathematical Society
  • 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
  • MathSciNet review: 3378837