Timeline for Classification of fibrations $\Bbb S^k\longrightarrow\Bbb S^d\longrightarrow B$
Current License: CC BY-SA 4.0
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| Nov 5, 2019 at 20:02 | comment | added | Igor Belegradek | Oh, apparently I looked at the version linked by F.C. whose first paragraph differs from the one you linked to which is why I was confused. In any case my point is very simple: The linked article of Ovsienko-Tabachnikov is not a reference for the fact that any fiber bundle with great sphere fibers is Hopf. Their article may contain such a reference but I do not immediately see it. On the other hand, the two articles by Browder I mentioned in comments do provide a reference (for a stronger claim). | |
| Nov 5, 2019 at 19:32 | history | edited | M. Winter | CC BY-SA 4.0 |
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| Nov 5, 2019 at 19:29 | comment | added | M. Winter | @IgorBelegradek I am honestly asking for your opinion: do you think that the linked article (more precisely: its first paragraph) at least claims that the projective spaces listed in the linked SE question are exactly the base spaces of spherical fibrations by great spheres? Because I do think so, and cannot understand how it can be interpreted otherwise. Do you further think that the answer by Neil is worth accepting. I feel not competent to judge its correctness. | |
| Nov 5, 2019 at 16:01 | comment | added | Igor Belegradek | The first paragraph in the linked paper says nothing on the matter. In my view the ideal answer is the one that gives a proof or an explicit reference to one. | |
| Nov 5, 2019 at 15:49 | comment | added | M. Winter | @IgorBelegradek I refer to the first paragraph of the linked paper. Yes, I do not ask for fibrations by great circles. But, I said in a comment that this was my actual intention. I am not changing the question now, since it seems to be (more) interesting as it is, and it has attracted a good answer (I do not fully understand it though, as this is not my field). I edited the "Update" insofar as I now talk about a "partial answer" which is more than satisfying for me. And without further explanation I do not understand how it is wrong. | |
| Nov 5, 2019 at 15:42 | comment | added | Igor Belegradek | I don't know what claim of Ovsienko-Tabachnikov you are referring to. In any case theirs is an expository article, i.e. not a primary reference with all proofs. In your question above you do not assume that the fibers are "great spheres". | |
| Nov 5, 2019 at 15:24 | history | edited | M. Winter | CC BY-SA 4.0 |
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| Nov 5, 2019 at 15:22 | comment | added | M. Winter | @IgorBelegradek I do not understand. The projective spaces for which the question at Math.SE was asked for are exactly the base spaces from above, as stated in "On fibrations with flat fibres" by Ovsienko and Tabachnikov, at least when we restrict to fibrations by great spheres. | |
| Nov 5, 2019 at 15:05 | comment | added | Igor Belegradek | What you linked to at stackexchange answers neither your question there, nor the one here, because in your case $B$ is simply-connected so that it equals to its universal cover. | |
| Nov 5, 2019 at 8:24 | history | edited | M. Winter | CC BY-SA 4.0 |
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| Nov 5, 2019 at 2:52 | comment | added | Igor Belegradek | On whether $B$ can admit a topological group structure, the answer is no. In fact, $B$ is not even an H-space. Indeed, theorem 6.10 in [Browder, W., Torsion in H-spaces. Ann. of Math. (2) 74 (1961), 24–51] says that if an H-space has finitely generated cohomology that vanish is all large degrees, then the first nonzero homotopy group of the space occurs in odd dimension. But $S^8$, $CP^m$, and homology $HP^m$ do not have this property. By Hurewicz theorem, their first nonzero homotopy groups occurs in degrees 8, 2, 4 respectively. | |
| Nov 4, 2019 at 23:26 | history | edited | M. Winter | CC BY-SA 4.0 |
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| Nov 3, 2019 at 21:05 | comment | added | M. Winter | @F.C. Thank you for hinting me to this very well written article! It seems to answer my question for the most part. Footnote 5 of this article reads "The next step would be a classification of affine Hopf fibrations. See [10, 19] for partial results. [...]". This suggests that the full question is still open. It turned out that I am actually already satisfied with a classification of fibrations by great circles, and this was achieved according to the paper "On fibrations with flat fibres.", also by Ovsienko and Tabachnikov. These are exactly the Hopf fibrations also found on Wikipedia. | |
| Nov 3, 2019 at 20:36 | comment | added | Ben McKay | This story is relevant to the study of the Blaschke conjecture in Riemannian geometry, as it provides some of the topological restrictions on Blaschke manifolds. | |
| Nov 3, 2019 at 18:18 | history | edited | YCor |
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| Nov 3, 2019 at 13:29 | answer | added | Neil Strickland | timeline score: 13 | |
| Nov 3, 2019 at 12:59 | history | edited | M. Winter | CC BY-SA 4.0 |
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| Nov 3, 2019 at 12:58 | comment | added | F. C. | A related reference is Ovsienko-Tabachnikov article on Hopf Fibrations and Hurwitz-Radon Numbers (ovsienko.perso.math.cnrs.fr/Publis/Hopf1.pdf). | |
| Nov 3, 2019 at 12:38 | history | asked | M. Winter | CC BY-SA 4.0 |