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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 3rd 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added some minimum content to _[[Stiefel manifold]]_, also a little bit to _[[Grassmannian]]_

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

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 3rd 2016
   • (edited May 3rd 2016)
   • PermaLink
   Author: Urs
   Format: MarkdownItexWhat'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](http://math.stackexchange.com/a/58074/58526) 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 3rd 2016
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   Author: DavidRoberts
   Format: MarkdownItexSchubert cells in flag manifolds are what I would look up. Don't know a reference, sadly.

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

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMay 3rd 2016
   • (edited May 3rd 2016)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThe cleanest account which I find is in this presentation, on the last slides * Zbigniew Błaszczyk, _On cell decompositions of $SO(n)$_ ([pdf](http://knm.mat.umk.pl/pliki/konferencje/zb_lsait3_2007.pdf))

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

   • Zbigniew Błaszczyk, On cell decompositions of $SO(n)$ (pdf)
5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 4th 2016
   • (edited May 4th 2016)
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   Author: DavidRoberts
   Format: MarkdownItexThere's a nice argument in the example on page 58 [in these notes](http://www3.nd.edu/~stolz/Math70330(F2008)/cohen.pdf) that the inclusion $V_n(\mathbb{C}^N) \to V_n(\mathbb{C}^{2N})$ is null-homotopic, for what it's worth.

   There’s a nice argument in the example on page 58 in these notes that the inclusion $V_n(\mathbb{C}^N) \to V_n(\mathbb{C}^{2N})$ is null-homotopic, for what it’s worth.

6. • CommentRowNumber6.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 4th 2016
   • (edited May 4th 2016)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexAlso, Theorem 3.3 in [this paper](http://web.stanford.edu/~kupers/liegroups.pdf) says that the inclusion of unitary groups $i\colon SU(k) \to SU(n)$, $k\lt n$, 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 $SU(k-m) \to SU(n-m)$, 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, $SU(n) \to SU(n)/SU(n-m)$ were cellular, we would probably be done. EDIT: Ah, I see page 301-302 of Hatcher's Algebraic Topology that the quotient map $SO(n) \to SO(n)/SO(n-k)$ 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](https://books.google.com.au/books?id=7FXtBwAAQBAJ&lpg=PA16&ots=ga7LQbHYrf&dq=quotient%20map%20stiefel%20manifold%20cellular&pg=PA13#v=onepage&q=quotient%20map%20stiefel%20manifold%20cellular&f=false) looks like it gives a cellular structure for the Stiefel manifold (there called $FQ(n,m)$) that would be compatible with inclusions.

   Also, Theorem 3.3 in this paper says that the inclusion of unitary groups $i\colon SU(k) \to SU(n)$, $k\lt n$, 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 $SU(k-m) \to SU(n-m)$, 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, $SU(n) \to SU(n)/SU(n-m)$ were cellular, we would probably be done.

   EDIT: Ah, I see page 301-302 of Hatcher’s Algebraic Topology that the quotient map $SO(n) \to SO(n)/SO(n-k)$ 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.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeMay 5th 2016
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   Author: Urs
   Format: MarkdownItexThanks!! I have recorded that [here](https://ncatlab.org/nlab/show/Stiefel+manifold#CellularInclusionOfStiefelManifolds). (No time for more at the moment.)

   Thanks!!

   I have recorded that here.

   (No time for more at the moment.)