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A $p$-compact group, as defined by Dwyer and Wilkerson, is a purely homotopically defined $p$-local analog of a compact Lie group. It has long been the hope, and later the conjecture, that these objects should have a classification similar to the classification of compact Lie groups. In this paper we finish the proof of this conjecture, for $p$ an odd prime, proving that there is a one-to-one correspondence between connected $p$-compact groups and finite reflection groups over the $p$-adic integers. We do this by providing the last, and rather intricate, piece, namely that the exceptional compact Lie groups are uniquely determined as $p$-compact groups by their Weyl groups seen as finite reflection groups over the $p$-adic integers. Our approach in fact gives a largely self-contained proof of the entire classification theorem for $p$ odd.
Kasper K. S. Andersen 1 ; Jesper Grodal 2 ; Jesper M. Møller 2 ; Antonio Viruel 3
@article{10_4007_annals_2008_167_95,
author = {Kasper K. S. Andersen and Jesper Grodal and Jesper M. M{\o}ller and Antonio Viruel},
title = {The classification of $p$-compact groups for $p$ odd},
journal = {Annals of mathematics},
pages = {95--210},
publisher = {mathdoc},
volume = {167},
number = {1},
year = {2008},
doi = {10.4007/annals.2008.167.95},
mrnumber = {2373153},
zbl = {1149.55011},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.167.95/}
}
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%0 Journal Article %A Kasper K. S. Andersen %A Jesper Grodal %A Jesper M. Møller %A Antonio Viruel %T The classification of $p$-compact groups for $p$ odd %J Annals of mathematics %D 2008 %P 95-210 %V 167 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.167.95/ %R 10.4007/annals.2008.167.95 %G en %F 10_4007_annals_2008_167_95
Kasper K. S. Andersen; Jesper Grodal; Jesper M. Møller; Antonio Viruel. The classification of $p$-compact groups for $p$ odd. Annals of mathematics, Tome 167 (2008) no. 1, pp. 95-210. doi: 10.4007/annals.2008.167.95
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