The torsion index of the spin groups
The torsion index of the spin groups
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自旋群的扭转指数
DOI：
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发表时间：
2005
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通讯作者：
B. Totaro
中科院分区：
文献类型：
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作者：
B. Totaro
挠指数是Grothendieck与任何连通紧李群G相关联的正整数[10]。如第1节所解释的，知道群的挠指数对G的交换子群、分类空间的积分上同调、分类空间的复协边、分类空间的Chow环[26]以及域上G-挠的分类有直接的影响[24]。Demazure [5]、马林[14]和Tits [25]已经给出了挠指数的上界。本文精确地计算了所有自旋群Spin（n）的挠指数。在另一篇文章中，我们将计算例外群E8的挠指数。如下所述，这就完成了所有单连通单群的挠指数的计算。随着维数的增加，自旋群的拓扑结构变得越来越复杂，因此我们根本不清楚是否有可能在所有维数下进行计算。事实上，答案是相当复杂的，并且在高维中的证明需要来自解析数论、鲍尔和班尼特关于λ 2的二进制展开的定理[1]的一些深层信息。
The torsion index is a positive integer associated by Grothendieck to any connected compact Lie group G [10]. As explained in section 1, knowing the torsion index of a group has direct consequences for the abelian subgroups of G, the integral cohomology of the classifying space, the complex cobordism of the classifying space, the Chow ring of the classifying space [26], and the classification of G-torsors over fields [24]. Demazure [5], Marlin [14], and Tits [25] have given upper bounds for the torsion index. In this paper, we compute the torsion index exactly for all the spin groups Spin(n). In another paper, we will compute the torsion index of the exceptional group E8. As discussed below, this completes the calculation of the torsion index for all the simply connected simple groups. The topology of the spin groups becomes more and more complicated as the dimension increases, and so it was not at all clear that it would be possible to do the calculation in all dimensions. Indeed, the answer is rather intricate, and the proof in high dimensions requires some deep information from analytic number theory, Bauer and Bennett’s theorem on the binary expansion of √ 2 [1].