On the third homology of SL_2 and weak homotopy invariance
On the third homology of SL_2 and weak homotopy invariance
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论SL_2的第三同调性与弱同伦不变性
DOI：
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发表时间：
2013
期刊：
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通讯作者：
Matthias Wendt
中科院分区：
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作者：
K. Hutchinson;Matthias Wendt
本文的目的是--在线性群SL_2的特殊情况下--对群同调与其A^1-不变置换之间的关系有一些了解。讨论了A^1-不变群同调的一些一般性质，如稳定化序列和Grothendieck-Witt模结构。结合关于精化Bloch群的非常精确的知识，这些方法可以推断，一般而言，群同调与其A^1不变版本之间存在相当大的差异。换言之，SL_2在许多非代数闭域上的弱同伦不变性是不成立的。
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 A^1-invariant replacement. We discuss some of the general properties of 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 to deduce that in general there is a rather large difference between group homology and its A^1-invariant version. In other words, weak homotopy invariance fails for SL_2 over many families of non-algebraically closed fields.