Timeline for "Abnormal" manifold
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7 events
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| Sep 4, 2016 at 10:04 | comment | added | André Henriques | Ok, here's a complete argument. Assume two disjoint opens. The boundary of an open defies a multi-valued vector field (for each point, a closed subset of the tangent line, bounded away from $0$ and from $\infty$). Take the max. That's a discontinuous everywhere non-zero vector field. Integrate to get an action of $\mathbb R$ on the long line. This action is faithful because the vector field is everywhere non-zero. Now apply the same argument as above. Contradiction. | |
| Sep 4, 2016 at 2:47 | comment | added | Greg Friedman | Thanks. I agree there's no everywhere nonzero vector field (this would also imply that the tangent bundle is trivial, which is known to be false: ams.org/journals/proc/1969-023-02/S0002-9939-1969-0246318-X/… ). And I agree that if there were such a vector field it would give open sets that separate the two sections mentioned. What I'm still not seeing is how not having such a vector field guarantees there's no way to put disjoint open sets around those sections. Or in other words, why would such open sets necessarily get you a nonvanishing vector field? | |
| Sep 2, 2016 at 23:11 | comment | added | André Henriques | An everywhere non-zero vector field would integrate to a faithful action of the reals on the long line. But that's impossible, as the limit as $t \to \infty$ of the flow applied to some given point $p$ cannot be long line's infinity, and must therefore be a fixed point of the action. | |
| Aug 9, 2016 at 4:34 | comment | added | Greg Friedman | Perhaps a dumb question: why not? (I realize this is an old MO question, but I just found it!) | |
| Nov 21, 2013 at 13:42 | vote | accept | Fujita Tomomi | ||
| Nov 21, 2013 at 13:37 | history | edited | André Henriques | CC BY-SA 3.0 |
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| Nov 21, 2013 at 13:18 | history | answered | André Henriques | CC BY-SA 3.0 |