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The classification of $ 2$-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
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Abstract: We prove that any connected $ 2$-compact group is classified by its $ 2$-adic root datum, and in particular the exotic $ 2$-compact group $ \operatorname{DI}(4)$, constructed by Dwyer-Wilkerson, is the only simple $ 2$-compact group not arising as the $ 2$-completion of a compact connected Lie group. Combined with our earlier work with Møller and Viruel for $ p$ odd, this establishes the full classification of $ p$-compact groups, stating that, up to isomorphism, there is a one-to-one correspondence between connected $ p$-compact groups and root data over the $ p$-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 $ p$-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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Additional Information:

Kasper K. S. Andersen
Affiliation: Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, Bygning 1530, DK-8000 Aarhus, Denmark
Email: kksa@imf.au.dk

Jesper Grodal
Affiliation: Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark
Email: jg@math.ku.dk

DOI: 10.1090/S0894-0347-08-00623-1
PII: S 0894-0347(08)00623-1
Received by editor(s): January 11, 2007
Posted: November 3, 2008
Additional Notes: The second author was partially supported by NSF grant DMS-0354633, an Alfred P. Sloan Research Fellowship, and the Danish Natural Science Research Council
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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