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ISSN 1088-6834(e) ISSN 0894-0347(p)

The classification of -compact groups

Author(s): Kasper K. S. Andersen; Jesper Grodal
Journal: J. Amer. Math. Soc. 22 (2009), 387-436.
MSC (2000): Primary 55R35; Secondary 55P35, 55R37
Posted: November 3, 2008
MathSciNet review: 2476779
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Abstract: We prove that any connected -compact group is classified by its -adic root datum, and in particular the exotic -compact group $\operatorname{DI}(4)$ , constructed by Dwyer-Wilkerson, is the only simple -compact group not arising as the -completion of a compact connected Lie group. Combined with our earlier work with Møller and Viruel for odd, this establishes the full classification of -compact groups, stating that, up to isomorphism, there is a one-to-one correspondence between connected -compact groups and root data over the -adic integers. As a consequence we prove the maximal torus conjecture, giving a one-to-one correspondence between compact Lie groups and finite loop spaces admitting a maximal torus. Our proof is a general induction on the dimension of the group, which works for all primes. It refines the Andersen-Grodal-Møller-Viruel methods by incorporating the theory of root data over the -adic integers, as developed by Dwyer-Wilkerson and the authors. Furthermore we devise a different way of dealing with the rigidification problem by utilizing obstruction groups calculated by Jackowski-McClure-Oliver in the early 1990s.

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