Timeline for Fundamental group of a topological group
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Apr 6, 2019 at 3:05 | review | Close votes | |||
| Apr 7, 2019 at 3:05 | |||||
| Mar 27, 2019 at 10:24 | answer | added | Neil Strickland | timeline score: 1 | |
| Mar 26, 2019 at 20:07 | answer | added | David White | timeline score: 6 | |
| Mar 26, 2019 at 19:38 | comment | added | YCor | There are examples of non-connected Lie groups $G$ for which it's not surjective: for instance, the quotient of the product of $\mathbf{R}$ and the integral Heisenberg group by identifying $\mathbf{Z}$ with the center of the latter. I don't know if such phenomena might be produced in connected, non-path-connected topological groups. | |
| Mar 26, 2019 at 19:34 | comment | added | YCor | What I would have liked to read is the observation that in the easiest examples, the resulting homomorphism is surjective, and not always injective. It's therefore natural to ask whether it's always surjective, and also in the restricted but important setting of connected Lie groups. Also, a side remark is that when $G$ is (possibly) not Hausdorff, the Hausdorffication quotient map $G\to G/_{\overline{\{1\}}}$ induces an isomorphism between $\pi_1$. In particular, looking at $G\to G/[G,G]$ or $G\to\overline{[G,G]}$ does not matter (there are connected Lie groups $G$ with $[G,G]$ not closed). | |
| Mar 26, 2019 at 19:28 | history | edited | David White | CC BY-SA 4.0 |
added 1 character in body
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| Mar 25, 2019 at 23:07 | comment | added | user74900 | @YCor I think in the setting of compact connected Lie groups, it can not happen that $\pi_1(Ab(G))$ is larger than $\pi_1G$. Can this fail for compact connected topological groups? | |
| Mar 25, 2019 at 21:04 | comment | added | David Roberts♦ | SU(2) and SO(3) are good ones... | |
| Mar 25, 2019 at 20:40 | review | Close votes | |||
| Mar 26, 2019 at 18:59 | |||||
| Mar 25, 2019 at 20:09 | comment | added | lab | @YCor no, do you have a trivial easy example ? | |
| Mar 25, 2019 at 20:06 | comment | added | YCor | Have you looked at any examples? | |
| Mar 25, 2019 at 19:59 | comment | added | lab | @YCor sure, how far the functorial homomorphism from being an isomorphism | |
| Mar 25, 2019 at 19:52 | comment | added | YCor | There's a functorial homomorphism. Have you looked at any examples? Do you have any more precise question? | |
| Mar 25, 2019 at 18:30 | review | First posts | |||
| Mar 25, 2019 at 20:24 | |||||
| Mar 25, 2019 at 18:28 | history | asked | lab | CC BY-SA 4.0 |