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nLab > Latest Changes: Stiefel manifold

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 3rd 2016

    I have added some minimum content to Stiefel manifold, also a little bit to Grassmannian

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 3rd 2016
    • (edited May 3rd 2016)

    What’s a good account of the CW-structure on Stiefel manifolds compatible with the CW-structure on the Grassmannians? Hatcher’s book discusses really just the latter. On Math.SE here somebody claims that I. M. James’ “The topology of Stiefel manifolds” is a “good reference for this sort of thing”. But there I only see some remarks on pages 5 and 21 of which I am not sure whether they really constitute a good reference for this sort of thing.

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 3rd 2016

    Schubert cells in flag manifolds are what I would look up. Don’t know a reference, sadly.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 3rd 2016
    • (edited May 3rd 2016)

    The cleanest account which I find is in this presentation, on the last slides

    • Zbigniew Błaszczyk, On cell decompositions of SO(n) (pdf)
    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 5th 2016
    • (edited May 5th 2016)

    There’s a nice argument in the example on page 58 in these notes that the inclusion Vn(ℂN)→Vn(ℂ2N) is null-homotopic, for what it’s worth.

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 5th 2016
    • (edited May 5th 2016)

    Also, Theorem 3.3 in this paper says that the inclusion of unitary groups i:SU(k)→SU(n), k<n, is cellular, and then 3.3.1 says that the action of SU(n−m) on SU(n) is cellular. I would be then happy to say that i is equivariant relative to the cellular map of groups SU(k−m)→SU(n−m), and so we have a chance for the inclusion of Stiefel manifolds to be cellular. I guess if we knew that the quotient map itself, SU(n)→SU(n)/SU(n−m) were cellular, we would probably be done.

    EDIT: Ah, I see page 301-302 of Hatcher’s Algebraic Topology that the quotient map SO(n)→SO(n)/SO(n−k) is cellular, for the cell structure on the Stiefel manifold inherited from the one on SO(n). I would guess this works for both complex and real, and ditto for my first paragraph.

    EDIT2: The following is probably not useful.

    Page 13-15 in The Topology of CW complexes looks like it gives a cellular structure for the Stiefel manifold (there called FQ(n,m)) that would be compatible with inclusions.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 5th 2016

    Thanks!!

    I have recorded that here.

    (No time for more at the moment.)

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 29th 2023

    Gave a proof of the corollary, linking to a result at compact object on the preservation of certain colimits by Top(Y,−) where Y is a compact space.

    diff, v12, current

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