Timeline for Is the tangent bundle of the long line $L$ homeomorphic to $L\times\mathbb R$?
Current License: CC BY-SA 2.5
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| Sep 11, 2010 at 19:01 | vote | accept | Zack | ||
| Aug 10, 2010 at 15:52 | comment | added | BS. | Indeed, this proves that they are not the same as topological fiber bundles (not only as vector bundles). As to general homeomorphism, I can only prove that this would imply the existence in $TL^+$ ($L^+$ the half ray) of an homeomorphic copy of $L^+$, disjoint from the zero section and with proper projection to $L^+$. I don't know how to exclude this. | |
| Aug 10, 2010 at 15:09 | comment | added | Chris Schommer-Pries | I think this second edit brings up an important point. When the OP says $L \times \mathbb{R}$ do they mean as a vector bundle (with the obvious action on the $\mathbb{R}$ factor)? or do they mean merely as a topological space? These are not obviously equivalent for the long line. So really it looks like there are two questions here. | |
| Aug 10, 2010 at 14:17 | history | edited | BS. | CC BY-SA 2.5 |
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| Aug 10, 2010 at 14:11 | history | edited | BS. | CC BY-SA 2.5 |
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| Aug 10, 2010 at 13:42 | history | answered | BS. | CC BY-SA 2.5 |