Timeline for Fundamental group as topological group
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 15, 2016 at 7:40 | history | edited | ACL | CC BY-SA 3.0 |
Correction of the mistake in the last sentence, as indicated by Georges Elencwajg and Jeremy Brazas
|
| Jun 15, 2016 at 7:38 | comment | added | ACL | @GeorgesElencwajg and Jeremy: Thank you for the clarification. | |
| Jun 14, 2016 at 8:58 | comment | added | Georges Elencwajg | Dear @Jeremy, you are absolutely right. Indeed ACL's last sentence is not what Bourbaki says. Bourbaki says that if $X$ is locally arcwise connected, then the adequate topology and the compact convergence topology have the same open subgroups. If Bourbaki knew that both topologies coincide he obviously would not have written this weaker statement! So you are perfectly right to be skeptical. And +1 for your expert answer, even I'm a bit late at the party :-) | |
| Jun 14, 2016 at 5:12 | comment | added | Jeremy Brazas | @GeorgesElencwajg Of course, everything is correct except for the last sentence. The "adequate" topology is a group topology by construction whereas the quotient topology is not always a topological group (see previous comments)..the two rarely coincide in non-discrete cases. Assuming $X$ is semilocally simply connected makes all three discrete. Perhaps this is what was intended. | |
| Jun 13, 2016 at 12:03 | comment | added | Georges Elencwajg | @Jeremy: ACL's answer is certainly correct since it comes from Bourbaki's Topologie Algébrique, Chapter III, pages 315, 316 and 335, where you can find proofs and more details. | |
| Jun 1, 2010 at 22:18 | comment | added | Jeremy Brazas | Additionally, the Hawaiian earring $\mathbb{HE}$ is locally arc connected but $\pi_{1}^{top}(\mathbb{HE})$ is not a topological group. So the group topology generated by adequate subgroups will not not coincide with with the quotient topology (which is not a group topology) even in some of the simplest non-discrete cases. These two other topologies are very interesting alternatives, I am just hesitant about some of the comparison statements. | |
| Jun 1, 2010 at 22:11 | comment | added | Jeremy Brazas | Yes, all three are discrete when a universal cover exists. Example 1 in the same paper is locally simply connected but not locally path connected. It has non-discrete topological fundamental group but the topology generated by admissible subgroups will be discrete. Thus the admissible topology is not always courser than the quotient topology. | |
| Jun 1, 2010 at 21:29 | comment | added | ACL | According to Fabel's Theorem 2 in Metric spaces with discrete topological fundamental groups, Topology and its Applications 154 (2007), 635-638, $\pi_1^{top}(X)$ is discrete for connected and locally path connected spaces that are semi-locally simply connected. In that case, all three topologies are discrete. | |
| Jun 1, 2010 at 19:21 | comment | added | Jeremy Brazas | It seems that for arc connected $X$, your admissible topology is discrete if and only if $X$ is semi-locally simply connected. But $\pi_{1}^{top}(X)$ is not always discrete for semi-locally simply connected $X$. This makes me think 1) and 2) are not always comparable. Am I missing something here? | |
| Jun 1, 2010 at 18:17 | history | answered | ACL | CC BY-SA 2.5 |