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.

Chevalleyp–local finite groups

“….q/ D BS 11 .q/ D BS 11 .u 6 / by Proposition 6.2 and Proposition A.10; the last equality follows from Proposition 6.5 since p .u 6 10 Chevalley p-local finite groups from Aguadé p-compact groups Write G i to denote one of the groups G 12 , G 29 , G 31 , or G 34 , and Z its center, namely,…”

Invariant Rings and Dual Representations of Dihedral Groups

“…So the 2-adic cohomology is also a polynomial algebra generated by elements of the same degree. We can find, [3], a compact connected Lie group G such that…”

Noetherian loop spaces

“…From Corollary 1.11 we obtain a classification of 3-connected p-Noetherian groups. They are given by the 3-connected covers of simply connected p-compact groups, which are known from the recent classification results ( [29], [30], [2], [3]). A general classification will be more difficult to obtain, even in the 2-connected case, as there are p-Noetherian groups fibering over a product of p-compact groups which do not split themselves as a product.…”

“…It is the "classifying space" BX that carries all of the information about the loop space. Amazingly enough, apart from compact Lie groups, there are only a few families of exotic p-compact groups and they have been recently completely classified by Andersen, Grodal, Møller, and Viruel: see [3] for the odd prime case and [2], [29], [30] for the prime 2 (the only exotic 2-compact group is basically the space DI (4) constructed by Dwyer and Wilkerson [16]). …”

“…To prove Theorem 4.1 one could also use the fact that p-compact groups satisfy the Peter-Weyl theorem (see [3,Theorem 1.6] and [2,Remark 7.3]). This would slightly shorten the proof and avoid the use of the maximal p-compact toral subgroup.…”