The classification of 2compact groups
Abstract
We prove that any connected 2compact group is classified by its 2adic root datum, and in particular the exotic 2compact group DI(4), constructed by DwyerWilkerson, is the only simple 2compact group not arising as the 2completion of a compact connected Lie group. Combined with our earlier work with Moeller and Viruel for p odd, this establishes the full classification of pcompact groups, stating that, up to isomorphism, there is a onetoone correspondence between connected pcompact groups and root data over the padic integers. As a consequence we prove the maximal torus conjecture, giving a onetoone 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 AndersenGrodalMoellerViruel methods to incorporate the theory of root data over the padic integers, as developed by DwyerWilkerson and the authors, and we show that certain occurring obstructions vanish, by relating them to obstruction groups calculated by JackowskiMcClureOliver in the early 1990s.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 November 2006
 arXiv:
 arXiv:math/0611437
 Bibcode:
 2006math.....11437A
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Group Theory;
 Primary: 55R35;
 Secondary: 55P35;
 55R37
 EPrint:
 47 pages