The torsion index of the spin groups. (English) Zbl 1094.57031
This paper is concerned with the torsion index, \(t(G)\), of a connected compact Lie group \(G\), as introduced by A. Grothendieck in the Séminaire C. Chevalley (Exposé 5, 1958). In [ibid., 219–248 (2005; Zbl 1093.57011)], the author gives the value of \(t(E_8)\). Here, he computes the torsion index exactly for all the spin groups, the torsion index being now known for all simply connected compact Lie groups. The paper begins with applications of the torsion index to complex cobordism, Chow rings, cohomology of classifying spaces and the classification of \(G\)-torsors over fields. The author uses the symplectic groups to give a lower bound for the torsion index of spin groups. The final part of the proof is a tour de force, obtained by inspecting the first 330 digits of \(\sqrt{2}\), using a theorem of M. Bauer and M. A. Bennet [Ramanujan J. 6, 209–270 (2002; Zbl 1010.11020)].
Reviewer: Daniel Tanré (Villeneuve d’ Ascq)
MSC:
| 57T10 | Homology and cohomology of Lie groups |
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