The classification of p-compact groups for p odd

• 成果类型: Review
• 署名作者: Andersen, K. K. S.; Grodal, J.; Moller, J. M.; Viruel, A.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2008.167.95
• 发表日期: 2008
• 页码: 95-210
• 关键词: classifying-spaces; homotopy classification; reflection groups; finite subgroups; higher limits; maximal tori; lie-groups; self-maps; realization; normalizers

摘要

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.