The classification of 2-compact groups

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The classification of 2-compact groups. / K. S. Andersen, Kasper; Grodal, Jesper.

I: Journal of the American Mathematical Society, Bind 22, Nr. 2, 2009, s. 387-436.

Publikation: Bidrag til tidsskriftTidsskriftartikelfagfællebedømt

Harvard

K. S. Andersen, K & Grodal, J 2009, 'The classification of 2-compact groups', Journal of the American Mathematical Society, bind 22, nr. 2, s. 387-436. https://doi.org/10.1090/S0894-0347-08-00623-1

APA

K. S. Andersen, K., & Grodal, J. (2009). The classification of 2-compact groups. Journal of the American Mathematical Society, 22(2), 387-436. https://doi.org/10.1090/S0894-0347-08-00623-1

Vancouver

K. S. Andersen K, Grodal J. The classification of 2-compact groups. Journal of the American Mathematical Society. 2009;22(2):387-436. https://doi.org/10.1090/S0894-0347-08-00623-1

Author

K. S. Andersen, Kasper ; Grodal, Jesper. / The classification of 2-compact groups. I: Journal of the American Mathematical Society. 2009 ; Bind 22, Nr. 2. s. 387-436.

Bibtex

@article{62f99ca017ec11de8478000ea68e967b,
title = "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 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 Moeller 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-Moeller-Viruel methods to incorporate the theory of root data over the p-adic integers, as developed by Dwyer-Wilkerson and the authors, and we show that certain occurring obstructions vanish, by relating them to obstruction groups calculated by Jackowski-McClure-Oliver in the early 1990s.",
author = "{K. S. Andersen}, Kasper and Jesper Grodal",
note = "Keywords: math.AT; math.GR; Primary: 55R35; Secondary: 55P35, 55R37",
year = "2009",
doi = "10.1090/S0894-0347-08-00623-1",
language = "English",
volume = "22",
pages = "387--436",
journal = "Journal of the American Mathematical Society",
issn = "0894-0347",
publisher = "American Mathematical Society",
number = "2",

}

RIS

TY - JOUR

T1 - The classification of 2-compact groups

AU - K. S. Andersen, Kasper

AU - Grodal, Jesper

N1 - Keywords: math.AT; math.GR; Primary: 55R35; Secondary: 55P35, 55R37

PY - 2009

Y1 - 2009

N2 - 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 Moeller 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-Moeller-Viruel methods to incorporate the theory of root data over the p-adic integers, as developed by Dwyer-Wilkerson and the authors, and we show that certain occurring obstructions vanish, by relating them to obstruction groups calculated by Jackowski-McClure-Oliver in the early 1990s.

AB - 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 Moeller 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-Moeller-Viruel methods to incorporate the theory of root data over the p-adic integers, as developed by Dwyer-Wilkerson and the authors, and we show that certain occurring obstructions vanish, by relating them to obstruction groups calculated by Jackowski-McClure-Oliver in the early 1990s.

U2 - 10.1090/S0894-0347-08-00623-1

DO - 10.1090/S0894-0347-08-00623-1

M3 - Journal article

VL - 22

SP - 387

EP - 436

JO - Journal of the American Mathematical Society

JF - Journal of the American Mathematical Society

SN - 0894-0347

IS - 2

ER -

ID: 11483385