Advanced

The classification of 2-compact groups

Andersen, Kasper LU and Grodal, Jesper (2009) In Journal of the American Mathematical Society 22(2). p.387-436
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 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... (More)
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 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. (Less)
Please use this url to cite or link to this publication:
author
publishing date
type
Contribution to journal
publication status
published
subject
in
Journal of the American Mathematical Society
volume
22
issue
2
pages
387 - 436
publisher
American Mathematical Society (AMS)
external identifiers
  • scopus:77950582963
ISSN
0894-0347
DOI
10.1090/S0894-0347-08-00623-1
language
English
LU publication?
no
id
3060e347-5d28-46ed-a329-17800df38bc9 (old id 3412526)
alternative location
http://www.ams.org/journals/jams/2009-22-02/S0894-0347-08-00623-1/
date added to LUP
2013-04-25 16:40:07
date last changed
2017-03-26 03:41:58
@article{3060e347-5d28-46ed-a329-17800df38bc9,
  abstract     = {We prove that any connected 2-compact group is classified by its<br/><br>
2-adic root datum, and in particular the exotic 2-compact group<br/><br>
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.},
  author       = {Andersen, Kasper and Grodal, Jesper},
  issn         = {0894-0347},
  language     = {eng},
  number       = {2},
  pages        = {387--436},
  publisher    = {American Mathematical Society (AMS)},
  series       = {Journal of the American Mathematical Society},
  title        = {The classification of 2-compact groups},
  url          = {http://dx.doi.org/10.1090/S0894-0347-08-00623-1},
  volume       = {22},
  year         = {2009},
}