Title: The classification of 2-compact groups
Authors: Kasper K. S. Andersen and Jesper Grodal
Emails: and
arXiv: math.AT/0611437
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 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
Moeller 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-Moeller-Viruel methods to
incorporate the theory of root data over the p-adic integers, as
developed by Dwyer-Wilkerson and the authors, and we show that certain
occurring obstructions vanish, by relating them to obstruction
groups calculated by Jackowski-McClure-Oliver in the early 1990s.