Timeline for Topological vs pro fundamental groups
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
|
|
| Feb 7, 2011 at 20:09 | comment | added | Jeremy Brazas | I see. Yes, you are certainly right in that case. | |
| Feb 7, 2011 at 14:10 | comment | added | Sergey Melikhov | Jeremy, thanks for clarifying: I of course meant not semi-$1$-LC but semi-LC$_1$, that is semi-locally simply-connected and locally connected. | |
| Feb 6, 2011 at 6:00 | comment | added | Jeremy Brazas | @Sergey Take the reduced suspension of the one point compactification of the natural numbers with isolated basepoint. This is a locally simply connected but non-locally path connected compact metric space. The quotient topology and the inverse limit topology on $\pi_1$ give non-discrete (but non-isomorphic since the first is not first countable) topological groups. | |
| Feb 6, 2011 at 0:23 | answer | added | Jeremy Brazas | timeline score: 1 | |
| Feb 5, 2011 at 23:56 | comment | added | Sergey Melikhov | Mike, for semi-locally simply-connected compact metrizable spaces $\pi_1(X)$ with either topology is discrete and is isomorphic to the inverse limit of the fundamental pro-group, and contains the same information as the fundamental pro-group. So indeed a "niceness condition trivializes the extra information" on both sides. Probably "compact" and "metrizable" are not essential restrictions here. I have no idea what is toposopic fundamental group, but for me the "well-behaved $\pi_1$" for non-locally connected spaces is the Steenrod $\pi_1$; see mathoverflow.net/questions/49526 | |
| Feb 5, 2011 at 22:45 | comment | added | Mike Shulman | @Theo JF: Thanks! I've edited the question along those lines to clarify what I'm specifically interested in at the moment. (-: | |
| Feb 5, 2011 at 22:44 | history | edited | Mike Shulman | CC BY-SA 2.5 |
clarified paths vs coverings
|
| Feb 5, 2011 at 22:18 | comment | added | Theo Johnson-Freyd | The toposophic fundamental group is (or, is roughly) the etale fundamental group. For badly behaved spaces this is a different and "better" group than the one describing maps from a circle. So the question has a lot of depth: the point is that various well-behavedness conditions on your space will lead to various relationships between the groups, but that (some of) these relationships should break in badly-behaved settings. I don't have any particular examples, just a philosophy, so this is a comment, not an answer. | |
| Feb 5, 2011 at 15:55 | answer | added | Sergey Melikhov | timeline score: 4 | |
| Feb 5, 2011 at 14:15 | comment | added | Sergey Melikhov | $\pi_1(X)$ also inherits a topology from the inverse limit topology on the Cech fundamental group; a base of this topology is given by the point inverses of all induced maps $f_*:\pi_1(X)\to\pi_1(P)$, where $P$ is a polyhedron and $f:X\to P$ is a map. With this topology, $\pi_1(X)$ is always a topological group. Its relationships with the fundamental pro-group of $X$ are rather well-understood (see e.g. arxiv.org/abs/0812.1407). I don't know though if the fundamental pro-group of $X$ (as defined e.g. in Dydak-Segal "Shape Theory") is the same as the pro-group you're interested in. | |
| Feb 5, 2011 at 9:19 | history | asked | Mike Shulman | CC BY-SA 2.5 |