The classification of $2$-compact groups

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.