## 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.## References

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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