The classification of $p$-compact groups for $p$ odd

Kasper K. S. Andersen¹ ; Jesper Grodal² ; Jesper M. Møller² ; Antonio Viruel³

¹ Department of Mathematical Science<br/>University of Aarhus <br/>8000 Aarhus C<br/>Denmark
² Department of Mathematical Science<br/>University of Copenhagen<br/>2100 Copenhagen<br/>Denmark
³ Departamento Álgebra, Geometría y Topología<br/>Universidad de Málaga<br/>29071 Málaga<br/>Spain

Annals of mathematics, Tome 167 (2008) no. 1, pp. 95-210.

Résumé

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.

Détail

DOI : 10.4007/annals.2008.167.95