The torsion index of E8 and other groups

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 ofG 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