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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 4th 2016
• (edited May 4th 2016)

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.

• CommentRowNumber6.
• CommentAuthorDavidRoberts
• CommentTimeMay 4th 2016
• (edited May 4th 2016)

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.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeMay 5th 2016

Thanks!!

I have recorded that here.

(No time for more at the moment.)