Timeline for Which groups admit a unique Lie group structure?
Current License: CC BY-SA 3.0
13 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| May 24, 2011 at 23:03 | comment | added | Ryan Budney | @algori: there are of course notions of manifold that do not demand 2nd countability. That was the point of my comment. | |
| May 24, 2011 at 21:53 | comment | added | algori | Ryan -- a Lie group is a smooth manifold, so it is second countable. | |
| May 24, 2011 at 21:48 | comment | added | Ryan Budney | If you don't demand Lie groups are 2nd countable, any abstract group with the discrete topology is a $0$-dimensional Lie group. I suspect this is what Claudio and Qiaochu are talking about. If you demand 2nd countability, these are not examples. | |
| May 24, 2011 at 21:46 | history | edited | algori | CC BY-SA 3.0 |
fixed the end of the argument
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| May 24, 2011 at 21:42 | comment | added | algori | Claudio -- I assume $G$ is a Lie group, so $G$ with discrete topology would not do. But you are right that in the end the argument uses connectedness. However, this is easily fixed. | |
| May 24, 2011 at 21:32 | comment | added | Qiaochu Yuan | Yes, I really think you want to talk about "unique connected Lie group structure" or else you can give any Lie group the discrete topology (making it a $0$-dimensional Lie group). | |
| May 24, 2011 at 21:28 | answer | added | Jim Humphreys | timeline score: 2 | |
| May 24, 2011 at 21:21 | comment | added | Claudio Gorodski | In 3., it seems you are assuming $G$ is compact and connected. For instance, you could have the discrete topology in $G$. | |
| May 24, 2011 at 20:37 | comment | added | algori | woops! Thanks, Ben, will correct this. | |
| May 24, 2011 at 20:35 | comment | added | Ben Webster♦ | Um, $T^n$ has $2^n$ elements of order 2. Not that that changes your point. | |
| May 24, 2011 at 20:34 | history | edited | algori | CC BY-SA 3.0 |
corrected a mistake
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| May 24, 2011 at 20:28 | history | asked | algori | CC BY-SA 3.0 |