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Is there anywhere a broad classification (or even a diagrammatic tree-structure representing) the topologies of Lie Groups?

Maybe this is a naïve question from a non-mathematician, but I would like to be able to visualise (at least partly, perhaps) the differences between the manifolds formed by group structures of interest in physics and chemistry, e.g. O(3) as a disconnected 2-part 3-manifold, SU(2) as a 3-sphere…

I’ve found Borel’s “TOPOLOGY OF LIE GROUPS AND CHARACTERISTIC CLASSES ” in which the Introduction states:

“a Lie group in the large is first a manifold, i.e., a topological Hausdorff space admitting a covering by open sets, each of which is homeomorphic to euclidian n-space; second it is a group; third it is a topological group,”…

… which has encouraged me to ask this question, and offer thanks for any helpful answers.

iSeeker
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    A broad classification would be along the lines of connected-disconnected, simply-connected-nonsimply connected, compact-noncompact. Among general results, you will find that each simply-connected Lie group is diffeomorphic to the product of some $R^n$ with some compact Lie group. Try to narrow down your question to make it answerable. – Moishe Kohan Aug 30 '20 at 18:00
  • Thanks for the start, Moishe. Can you, or anyone else suggest references where I might find chunks of those properties already tabulated (rather than trawling through wikipedia or Fulton & Harris)? – iSeeker Aug 30 '20 at 18:38
  • For instance https://math.stackexchange.com/questions/16779/conditions-for-a-smooth-manifold-to-admit-the-structure-of-a-lie-group – Moishe Kohan Aug 30 '20 at 18:41
  • Just checked that and refs therein - I was hoping to find tabulations naming specific groups such as https://en.wikipedia.org/wiki/Compact_group which lists e.g.
    1. the circle group T and the torus groups Tn, 2) the orthogonal groups O(n), 3) the special orthogonal group SO(n) and its covering spin group Spin(n), 4) the unitary group U(n) and the special unitary group SU(n), Perhaps I'm expecting too much. Looks like I'd better pursue the suggestion in your first comment.
     – iSeeker Aug 30 '20 at 18:51

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