Timeline for For unit sphere bundle over sphere there exist a real vector bundle equipped with inner product structure? [closed]
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 16, 2023 at 8:18 | history | closed |
user44191 Ryan Budney Stefan Waldmann Max Horn Dave Benson |
Needs details or clarity | |
| Sep 15, 2023 at 13:04 | comment | added | Sebastian Goette | One can construct smooth bundles of spheres over spheres with structure group the diffeomorphism group of the fibre where the statement is false in the smooth category. In particular, these bundles are not the unit sphere bundles of smooth Euclidean vector bundles. See Igusa's paper Axioms for higher torsion invariants of smooth bundles. The base should be $S^{4k}$ and the fibre an odd sphere of some very large dimension (like $12k+7$ or so, I don't remember). | |
| Sep 13, 2023 at 9:19 | comment | added | Dimpi Paul | @ Thomas Rot You mean only vector bundle with fiber S^1 or S^2 and whatever the base space this statement is true? | |
| Sep 13, 2023 at 9:15 | comment | added | Dimpi Paul | @ Nicolast right exactly. | |
| Sep 13, 2023 at 8:26 | comment | added | Thomas Rot | There is a satisfying answer in terms of classifying spaces which depends on the dimension of the sphere. I think it is false for high dimensional spheres, but I think this is true for S^1 and S^2 bundles as Diff(S^1) is homotopy equivalent to O(2) and Diff(S^2) is homotopy equivalent to O(3). | |
| Sep 13, 2023 at 7:40 | comment | added | Nicolast | Ok I guess I understand your question now: you want to know whether any topological (smooth ?) sphere bundle over manifold (or something else ?) is homeomorphic (diffeomorphic ?) to the unit sphere bundle of a vector bundle with an inner product. Right ? Could you please rephrase your question as it is quite obscure at the moment ? | |
| Sep 13, 2023 at 6:47 | comment | added | Dimpi Paul | So, i am very curious to know that: this is true in general or with some restrictions? For a sphere bundle over sphere, we get an n-plane bundle which just an extension of given sphere bundle? | |
| Sep 13, 2023 at 6:36 | comment | added | Dimpi Paul | @ David Actually i have seen in some paper that for the bundle say A: S^ 2 -> CP^3 -> S^4 , there exists a 3-plane bundle over S^4 say B, such that A is restriction of B on unit sphere. | |
| Sep 13, 2023 at 6:23 | comment | added | Dimpi Paul | @Nicolast any vector bundle over compact Hausdroff base space admit inner product structure. | |
| Sep 13, 2023 at 1:34 | comment | added | David Roberts♦ | We're you thinking of an associated bundle? You can let $O(n+1)$ act on the fibres (assuming the structure group of the fibre bundle is an orthogonal group) and by the defining representation on $\mathbb{R}^{n+1}$, and make the associated vector bundle. If the structure group is homeomorphisms of a sphere, its not clear to me it can be done. | |
| Sep 12, 2023 at 20:09 | review | Close votes | |||
| Sep 16, 2023 at 8:18 | |||||
| Sep 12, 2023 at 20:05 | comment | added | Ben McKay | When you say unit sphere bundle, how is that different from a topological fiber bundle with fibers homeomorphic to spheres? In other words, what is the significance of the word unit? | |
| Sep 12, 2023 at 19:48 | comment | added | Nicolast | Could you clarify your question ? Any vector bundle over any manifold admits an inner product... | |
| S Sep 12, 2023 at 19:17 | review | First questions | |||
| Sep 12, 2023 at 19:49 | |||||
| S Sep 12, 2023 at 19:17 | history | asked | Dimpi Paul | CC BY-SA 4.0 |