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nLab > Latest Changes: compact Lie group

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 21st 2020

    added

    • more hyperlinks (and some whitespace) to the paragraph on maximal tori.

    • the statement that smooth actions of compact Lie groups on smooth manifolds are proper

    diff, v9, current

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 24th 2020

    Corrected the reference: it’s Corollary 21.6 in the actual second edition.

    diff, v10, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 24th 2020

    Ah, thanks.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 13th 2021

    added a section “Applications – In equivariant homotopy theory” (here) with this remark:

    Compact Lie groups make a somewhat unexpected appearance as equivariance groups in equivariant homotopy theory, where the compact Lie condition on the equivariance group is needed in order for (the available proofs of) the equivariant Whitehead theorem to hold.

    diff, v12, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 13th 2021

    added brief mentioning of the equivariant triangulation theorem under “Properties” (here)

    diff, v12, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeApr 8th 2021

    added pointer to:

    diff, v15, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeApr 8th 2021

    added brief mentioning (here) that every compact Lie group admits a bi-invariant metric

    diff, v16, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeNov 4th 2021

    I have added (here) the statement that connected compact abelian Lie groups are tori (with reference to Adams 1982) and then statement and proof that compact abelian Lie groups are direct products of tori with finite abelian groups.

    For ease of cross-linking this at other relevant pages, I’ll give this also it’s own little stand-alone page “classification of abelian compact Lie groups”.

    diff, v22, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMar 20th 2023

    added pointer to:

    diff, v23, current

1 to 9 of 9

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