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:
https://lup.lub.lu.se/record/3412526
- author
- Andersen, Kasper
LU
and Grodal, Jesper
- publishing date
- 2009
- 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
- 2016-04-01 12:30:06
- date last changed
- 2022-03-29 01:46:24
@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}},
doi = {{10.1090/S0894-0347-08-00623-1}},
volume = {{22}},
year = {{2009}},
}