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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

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 7th 2025

    Typo in references.

    Also, I think this proposition:

    The Stiefel manifold Vn(k) is (n-1)-connected.

    should have some bounds on n and k. For low dimensions I think it may not be true, particularly around the region of the the non-simple Spin(4).

    diff, v15, current

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 7th 2025

    Seems like it should be (k−n−1)-connected. There’s a proof in the answer here. I’ll see if I can see what’s wrong with our proof.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJul 7th 2025

    Thanks for the alert. Was busy all morning but looking into it now.

    Seems to me that the entry is locally correct, but globally the definitions of k and n don’t match across paragraphs. Editing now…

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJul 7th 2025

    fixed the variable declaration mismatch in this prop.

    In addition I replaced “k” throughout by “N” to make it easier to remember that this is the “large one of the two dimensions”. (This wasn’t quite the source of the previous mistake, which was instead switching between n and N−n=k−n, but it is still the naturally more suggestive naming of the variables.)

    also made various other small cosmetic adjustments to the source code.

    diff, v16, current

    • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 7th 2025
    • (edited Jul 7th 2025)

    [Deleted]

    • CommentRowNumber14.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 8th 2025

    The degrees of homotopy groups in the proof didn’t actually match the statement in the proposition—the former still implicitly referred to (n−1)-connectedness. There were also still inconsistent labelling of dimensions and so on.

    I have made all the ns into ks and then the citation to the connectedness of the inclusion map of orthogonal groups, in terms of n, can be extracted cleanly without contortions.

    I don’t know what this paragraph is doing in this article:

    According to Yokota 1956, the inclusions SU(k)↪SU(N) are cellular such that this is compatible with the group action (reviewed here in 3.3 and 3.3.1). This implies that also the projection SU(N)→SU(N)/SU(N−k) is cellular (e.g. Hatcher 2002 p. 302).

    and we should probably put this on an appropriate separate page.

    I also added a note:

    More generally, given any finite-dimensional inner product space (V,⟨,⟩), there is a corresponding orthogonal group O(V), and one can repeat the above definition more or less verbatim.

    I didn’t change the label on the definition that is EOn, in case that broke stuff elsewhere on the nLab.

    diff, v17, current

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