Timeline for Is every real vector bundle over the circle necessarily trivial?
Current License: CC BY-SA 2.5
6 events
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| Jan 8 at 7:33 | comment | added | horned-sphere | This answer is also illuminating for distinguishing a line bundle from an embedding of it in $\mathbb{R}^3$ (there's infinitely distinct many), which tripped me at first. math.stackexchange.com/questions/1100871/… | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Sep 8, 2010 at 7:58 | vote | accept | Orbicular | ||
| Sep 8, 2010 at 2:40 | comment | added | Victor Protsak | More generally, $G$-bundles on $S^n$ are classified by the elements of $\pi_{n-1}(G):$ after trivializing over two enlarged hemispheres, the transition function becomes a map from the equatorial $S^{n-1}$ into $G.$ | |
| Sep 7, 2010 at 21:14 | comment | added | Paul Yuryev | For example, for (real) line bundles this is very easy: without loss of generality we may assume that local trivilialization charts are $(0,1)\times \mathbb{R}^1$ and that the transition functions are $\pm 1$. Then by going over the circle counterclockwise we may "fix" all transition functions to be 1 (by changing the local trivialization frame $e\to-e$ if need be), except possibly for the last one -- thus there are only two line bundles. | |
| Sep 7, 2010 at 19:24 | history | answered | Dan Ramras | CC BY-SA 2.5 |