The torsion index of$E_8$ and other groups
The torsion index of$E_8$ and other groups
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$E_8$等组的扭转指数
DOI：
10.1215/s0012-7094-05-12922-2
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发表时间：
2005
影响因子：
2.5
通讯作者：
B. Totaro
中科院分区：
文献类型：
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作者：
B. Totaro
扭转指标是由Grothendieck与任意连通紧李群G[10]关联的正整数。群的扭转指数的确定对分类空间BG的积分上同调、BG的复配、BG的Chow环以及g -环在场上的分类都有直接的影响。我的论文[25]第1节对扭转指标的这些应用进行了总结。迄今为止，扭转指数的最佳上界是Tits[21]。在本文中，我们特别证明了例外群E8的扭转指数等于26325 = 2880。与我关于自旋群[25]的论文一起，完成了所有单连通紧李群的扭转指标的计算。本文还计算了PSO（2n）、E6/(Z/3)和E7/(Z/2)群的扭转指标，从而完成了所有伴随型紧李群的扭转指标的计算。E8的扭转指标的计算意味着我的结果的最优性，即在一个场上的每个E8-扭转量在除以26325[24]的某个程度的场扩展上变得微不足道。（等价地，域上的每个类型为E8的代数群都被划分为某个除26325次的域扩展。）实际上，根据Grothendieck关于挠性指标[10]的定理和本文的计算，存在一个域k和一个E8-torsor X在k上，使得X在其上变得平凡的任何有限扩展域的度数都是k的26325倍。到目前为止，对这个数字的最好估计是它必须在223·5 = 60和29335 = 69120之间，由Tits[21]。素数p除以G的扭转指标恰好是使分类空间BG的积分上同调具有p-扭转的素数，或者等价地使G本身的积分上同调具有p-扭转的素数。这些“扭转素数”对于所有紧李群都是已知的，Borel在1961年给出了最终的答案：它们是G的基本群的扭转子群的阶数的素数，如果G的全称覆盖有一个简单因子Spin(n)类型，对于n≥7或在F4， E6和E7中有G2, 2和3，以及在E8中有2,3和5。为定义扭转指标，设T为紧李群G中的极大环面，设N为标志流形G/T的复维数。环面T的每个特征决定了G/T上的复线束。考虑由这些线束的H2（G/T，Z）中的Chern类生成的G/T的积分上同调的子带。然后定义G的扭转指标为最小的正整数t(G)，使得t(G)乘以H2N （G/ t，Z）中一个点的类属于这个子带。让我总结一下关于扭转指标的计算。也许最重要的早期结果，尽管它的表述有些不同，是用子群的扭转指标表示的扭转指标的上界
The torsion index is a positive integer associated by Grothendieck to any connected compact Lie group G [10]. Knowing the torsion index of a group has direct consequences for the integral cohomology of the classifying space BG, the complex cobordism of BG, the Chow ring of BG, and the classification of G-torsors over fields. These applications of the torsion index are summarized in my paper [25], section 1. The best upper bounds for the torsion index so far have been those of Tits [21]. In this paper, we show in particular that the exceptional group E8 has torsion index equal to 26325 = 2880. Together with my paper on the spin groups [25], this completes the calculation of the torsion index for all simply connected compact Lie groups. We also compute the torsion index of the groups PSO(2n), E6/(Z/3), and E7/(Z/2) in this paper, which completes the calculation of the torsion index for all compact Lie groups of adjoint type. The calculation of the torsion index of E8 implies the optimality of my result that every E8-torsor over a field becomes trivial over some field extension of degree dividing 26325 [24]. (Equivalently, every algebraic group of type E8 over a field becomes split over some field extension of degree dividing 26325.) Indeed, by Grothendieck’s theorem on the torsion index [10] together with this paper’s calculation, there is a field k and an E8-torsor X over k such that any finite extension field over which X becomes trivial has degree a multiple of 26325 over k. Until now the best estimates of this number were that it must be between 223 · 5 = 60 and 29335 = 69120, by Tits [21]. The prime numbers p dividing the torsion index of G are precisely those such that the integral cohomology of the classifying space BG has p-torsion, or equivalently those such that the integral cohomology of G itself has p-torsion. These “torsion primes” are known for all compact Lie groups, the final answer being given by Borel in 1961 [5]: they are the primes dividing the order of the torsion subgroup of the fundamental group of G, together with 2 if the universal cover of G has a simple factor of type Spin(n) for n ≥ 7 or G2, 2 and 3 in the cases F4, E6, and E7, and 2, 3, and 5 in the case E8. To define the torsion index, let T be a maximal torus in a compact Lie group G, and let N be the complex dimension of the flag manifold G/T . Each character of the torus T determines a complex line bundle on G/T . Consider the subring of the integral cohomology of G/T generated by the Chern classes in H2(G/T,Z) of these line bundles. Then the torsion index of G is defined as the smallest positive integer t(G) such that t(G) times the class of a point in H2N (G/T,Z) belongs to this subring. Let me sum up the calculations that have been made of the torsion index. Probably the most important early result, although it was stated somewhat differently, is an upper bound for the torsion index in terms of the torsion index of a subgroup