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nLab > Latest Changes: BO(n)

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  1. Created page for BO(n), the classifying space of the orthogonal group O(n). (See discussion on Stiefel-Whitney class.) There is still a lot to add though.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorSamuel Adrian Antz
    • CommentTimeMar 10th 2024
    • (edited Mar 10th 2024)

    Edit: Accidently pushed the same message again. There seem to be some render issues needing to be fixed. Edit: They are fixed now.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 11th 2024

    Good.

    In the first sentence I have added a parenthetical making it read:

    BO(n)B O(n) is the classifying space for (principal bundles with structure group) the orthogonal group O(n)O(n).

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 11th 2024

    added cross-link with KO

    diff, v2, current

  5. Changed “limit” to “colimit” and added stable orthogonal group like on BU(n).

    diff, v3, current

    • CommentRowNumber6.
    • CommentAuthorSamuel Adrian Antz
    • CommentTimeMar 12th 2024
    • (edited Mar 12th 2024)

    Added the simplest classifying space BO(1)≅ℝP ∞B O(1)\cong\mathbb{R}P^\infty and its classification of O(1)O(1) principal bundles. (A section about the general classification will follow soon.) Also linked the Wikipedia page Classifying space for O(n).

    Edit: As a link including “)” can’t be linked due to disrupting the

    [….](….)

    environment, I have created a redirect from “Classifying space for orthogonal group”.

    diff, v4, current

  7. Added reference for definition in Milnor & Stasheff. Added connection to characteristic classes. (I plan to create the page for balanced product in the future.) Also linked pages of BSO(n) and BSU(n), which I will create either today or tomorrow.

    diff, v5, current

  8. The classification result below is for principal O(1)-bundles, not principal U(1)-bundles.

    AD

    diff, v6, current

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