The classification of 2-compact groups

Abstract

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