The classification of $2$-compact groups
Authors:
Kasper K. S. Andersen and Jesper Grodal
Journal:
J. Amer. Math. Soc. 22 (2009), 387-436
MSC (2000):
Primary 55R35; Secondary 55P35, 55R37
DOI:
https://doi.org/10.1090/S0894-0347-08-00623-1
Published electronically:
November 3, 2008
MathSciNet review:
2476779
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Abstract: We prove that any connected $2$âcompact group is classified by its $2$âadic root datum, and in particular the exotic $2$âcompact group $\operatorname {DI}(4)$, constructed by DwyerâWilkerson, is the only simple $2$âcompact group not arising as the $2$âcompletion of a compact connected Lie group. Combined with our earlier work with MĂžller and Viruel for $p$ odd, this establishes the full classification of $p$âcompact groups, stating that, up to isomorphism, there is a one-to-one correspondence between connected $p$âcompact groups and root data over the $p$âadic 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 AndersenâGrodalâMĂžllerâViruel methods by incorporating the theory of root data over the $p$âadic integers, as developed by DwyerâWilkerson and the authors. Furthermore we devise a different way of dealing with the rigidification problem by utilizing obstruction groups calculated by JackowskiâMcClureâOliver 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
Received by editor(s):
January 11, 2007
Published electronically:
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
Article copyright:
© Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.