Timeline for Topological vs pro fundamental groups
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 22, 2022 at 7:16 | history | edited | CommunityBot |
replaced http://front.math.ucdavis.edu/ with https://arxiv.org/abs/
|
|
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
|
|
| Feb 7, 2011 at 17:47 | history | edited | Sergey Melikhov | CC BY-SA 2.5 |
added 11 characters in body
|
| Feb 7, 2011 at 17:42 | history | edited | Sergey Melikhov | CC BY-SA 2.5 |
added 2565 characters in body
|
| Feb 7, 2011 at 15:00 | history | edited | Sergey Melikhov | CC BY-SA 2.5 |
added 2426 characters in body
|
| Feb 7, 2011 at 5:48 | comment | added | Mike Shulman | I'm personally inclined to believe in the fundamental pro-group as "the right thing to consider" for abstract reasons, but I'd hoped to hear some concrete reasons either for that or against it, e.g. in terms of what properties of spaces each can distinguish. It's not a priori obvious to me (nor is it, apparently, to the people who study it) that the quotient topology is wrong just because it's not always a topological group. | |
| Feb 6, 2011 at 4:03 | history | edited | Sergey Melikhov | CC BY-SA 2.5 |
added 1960 characters in body
|
| Feb 6, 2011 at 1:36 | comment | added | Jeremy Brazas | The quotient topology on $\pi_1$ is finer (in many cases strictly finer) than the one Sergey is talking about by the universal property of quotient spaces. | |
| Feb 6, 2011 at 0:14 | comment | added | Sergey Melikhov | Regarding the two topologies on $\pi_1$: I've looked at some examples a couple years ago, and unfortunately I can't remember anything now and I'm kind of short of time packing up for a flight tomorrow morning. Frankly speaking, I don't know any good reason to consider the quotient topology (other than it being easy to define). The "inverse limit" topology agrees with the group structure and its Hausdorff quotient is metrizable as long as $X$ is compact metrizable. Nothing like this holds for the quotient topology. | |
| Feb 5, 2011 at 23:35 | comment | added | Sergey Melikhov | Sorry, I meant this only for pro-groups indexed by the positive integers. I should have said that I'm only really addressing the case where $X$ is a compact metrizable space. For such $X$ the fundamental pro-group is isomorphic to one indexed by the positive integers. | |
| Feb 5, 2011 at 23:30 | comment | added | Mike Shulman | If you can say anything about why and when the inverse-limit topology is "often the same" as the quotient topology, though, then I think that together with your remarks above (which I have yet to digest) that would go a ways towards answering the original question. | |
| Feb 5, 2011 at 23:03 | comment | added | Mike Shulman | Thanks, this is interesting, although not really the question I was asking since the topology is different. Is it really true that the inverse-limit topological group contains all the information of the fundamental pro-group if the latter is Mittag-Leffler? I'm pretty sure it's not true for the inverse-limit topological group of an arbitrary pro-group; there's a counterexample due to Higman & Stone which I summarized here. | |
| Feb 5, 2011 at 15:55 | history | answered | Sergey Melikhov | CC BY-SA 2.5 |