The Topology of Lie Groups: a Collection of Results

This article summarizes some results about the global structure (topology) of Lie groups, including their homotopy groups and homology groups.

Revision history

2025-01-26 (added a reference)
2024-12-08 (updated bibliography)
2024-11-27 (added a footnote about second cohomology groups)
2024-11-10 (made the language in the section about Lie algebras more precise by replacing "same" with "isomorphic" where appropriate)
2024-05-21 (changed the title to be more descriptive, made the abstract more descriptive)
2024-05-19 (added a reference)
2024-04-20 (changed equals to isomorphic in one equation, added a note about Bott periodicity)
2024-04-16 (added another definition of semisimple for compact connected Lie groups, added a footnote defining semisimple for not-necessarily-compact Lie groups, added a section with examples for a general structure theorem, added a section about maximal tori and rank, added a sentence relating the rank to the notation, added sections about homology groups and torsion, added a section about cohomology rings and their relationship to homology groups, changed "group" to "compact topological group" in the footnote about Tannaka-Krein duality and removed an unhelpful reference)
2024-04-08 (first version)

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updated 2025-02-25