Timeline for Is there a good notion of morphism between orbifolds?
Current License: CC BY-SA 3.0
27 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
|
|
| Dec 23, 2013 at 14:03 | history | edited | André Henriques | CC BY-SA 3.0 |
added 223 characters in body
|
| Dec 16, 2013 at 19:29 | comment | added | André Henriques | @Bruno. You are absolutely correct. But in your question you explicitly asked for a notion of maps from [0,1] into an orbifold that captures the notion of fundamental group. Now it's no longer a just a matter of taste: with one approach things do work out nicely, and with another approach things simply do not work out. | |
| Dec 15, 2013 at 21:44 | comment | added | Bruno Martelli | Just a comment: the "importance" of a definition depends on taste and on the objects you are interested in today. If you want to study hyperbolic or Seifert 3-manifolds and 3-orbifolds, you might not feel the need of allowing non-effective actions or some other kinds of generalizations (AFAIK the most important papers on 3-orbifolds don't treat this class.) Of course if you are interested in moduli space things change a lot. Any approach or different viewpoint is interesting, but I wouldn't say that there is a "best" one (we are indebted to Thurston also for this way of seeing mathematics...) | |
| Dec 15, 2013 at 21:35 | comment | added | HJRW | Well, unless there's some further issue lurking, it certainly seems to be possible to 'follow your nose' and get the right definition without thinking about 2-categories, even if that definition can be described in those terms. | |
| Dec 15, 2013 at 20:44 | comment | added | André Henriques | It is indeed difficult to convince people that one does need to use 2-categories in order to get a definition that allows for a definition of the fundamental group via paths and homotopies. But I remain firmly convinced that this is indeed the case. I'll point out that your proposal "...a compatible choice of local morphisms $\hat f$ at each point..." already has a little bit of a 2-categorical flavor, as you might want to declare some of those choices equivalent in order not to get too many different morphisms between orbifolds. | |
| Dec 15, 2013 at 20:22 | comment | added | HJRW | ... I still believe this is essentially the definition Thurston would have used (cf. Mike Davies' comments referred to above). I also don't see why one needs to worry about 2-categories in order to get this definition right. | |
| Dec 15, 2013 at 20:20 | comment | added | HJRW | Sorry, I misread the Kleiner--Lott definition. It seems to me that their definition is the wrong one (though very close to the right one, and perhaps it works well for their purposes). I would prefer to say that a morphism of orbifolds is precisely a compatible choice of local morphisms $\hat{f}$ at each point (where $\hat{f}$ is as in their definition). With this definition, path lifting and homotopy lifting both go through. As you say, there are some uniqueness issues, but these are morally equivalent to the choice that one makes when choosing a base point... (cont'd) | |
| Dec 15, 2013 at 14:26 | comment | added | André Henriques | @HJRW: The uniquness part of path lifting fails. And homotopy lifting fails all together. | |
| Dec 15, 2013 at 7:48 | comment | added | HJRW | To clarify, I just meant that Thurston would have performed the computation similarly. | |
| Dec 15, 2013 at 7:04 | comment | added | HJRW | ... I don't know whether I'm working in a 1-category, a 2-category, or an $\infty$-category with bells on, but this is what you get by following your nose, and I'm pretty sure Thurston would have said something similar (cf. Claudio Gorodski's comment on Patrick I-Z's answer). | |
| Dec 15, 2013 at 7:03 | comment | added | HJRW | I don't think there's any doubt that the Kleiner--Lott definition is the one Thurston would have given. There's an obvious generalization of the notion of covering map to this context, and one can prove the path-lifting and homotopy-lifting lemmas, whence covering maps are (orbifold) $\pi_1$-injective. The quotient map of $S^1$ that you mention is an orbifold covering map, so it follows that the quotient orbifold has infinite fundamental group (and indeed one can easily see that it's $D_\infty$)... (cont'd) | |
| Dec 15, 2013 at 0:27 | history | edited | André Henriques | CC BY-SA 3.0 |
deleted 30 characters in body
|
| Dec 14, 2013 at 23:01 | comment | added | André Henriques | @HJRW. It's not absurd. It's just the way it is. If you force orbifolds to be a category (as opposed to a 2-category), and if you define fundamental groups using paths and homotopies, then you don't get what you want. You get the fundamental group of the coarse moduli space. | |
| Dec 14, 2013 at 21:21 | comment | added | HJRW | I don't really understand what you're attributing to whom, but the implication that Thurston's definition naturally leads to an incorrect computation of the orbifold fundamental group is absurd. This just means you didn't 'follow your nose' correctly. | |
| Dec 14, 2013 at 19:58 | history | edited | André Henriques | CC BY-SA 3.0 |
added 36 characters in body
|
| Dec 14, 2013 at 19:48 | history | edited | André Henriques | CC BY-SA 3.0 |
added 36 characters in body
|
| Dec 14, 2013 at 19:44 | comment | added | André Henriques | @Qfwfq: Yes. They correspond to gerbes. If you allow me to use the analogy bundles, then the gerbe is the bundle (i.e. the triple consisting of a total space, a base space, and a map between them), whereas the purely ineffective orbifold is just the total space. Of course, one can recover the base space from the total space (by taking the coarse moduli space), so there is really no difference here. The only thing is that one can consider gerbes over things other than manifolds. E.g. one can consider gerbes over orbifolds (which could themselves be non-effective). | |
| Dec 14, 2013 at 16:59 | comment | added | André Henriques | Using the correct definition of morphisms between orbifolds (i.e. maps between stacks), one can define the fundamental group in the usual way (with paths and homotopies), and one gets the same thing as for the definition via covering spaces. | |
| Dec 14, 2013 at 16:26 | comment | added | Lee Mosher | ... "An orbifold $O$ is a space locally modelled on $R^n$ modulo finite group actions. Here is the formal definition: $O$ consists of a Hausdorff space $X_O$, with some additional structure. $X_O$ is to have a covering by a collection of open sets ${U_i}$ closed under finite intersections. To each $U_i$ is associated a finite group $\Gamma_i$, an action of $\Gamma_i$ on an open subset $\tilde U_i$ of $R^n$ and a homeomorphism $\phi_i : U_i \approx \tilde U_i / \Gamma_i$..." and so on, describing the compatability condition when $U_i \subset U_j$... | |
| Dec 14, 2013 at 16:23 | comment | added | Lee Mosher | @Andre: You have, by truncation, misquoted Thurston. He is employing a rhetorical device used by many of us: first give tne intuition, then give the formalities, which he does immediately. Here is a fuller quote from Section 13.2 of "The Geometry and Topology of Three-Manifolds": | |
| Dec 14, 2013 at 9:39 | comment | added | Bruno Martelli | always AFAIK, with Thurston's definition the fundamental group is the deck transformation group of the universal orbifold covering, and is hence $D_\infty$ with $S^1/Z_2$. | |
| Dec 13, 2013 at 23:44 | comment | added | Qfwfq | Do what you call "purely ineffective orbifolds" correspond to "(so and so) gerbes"? That is, something locally isomorphic to $U\times \mathrm{B} G$, where $\mathrm{B}G:=[\textrm{point}//G]$. | |
| Dec 13, 2013 at 21:13 | history | edited | André Henriques | CC BY-SA 3.0 |
added 2531 characters in body
|
| Dec 13, 2013 at 16:39 | comment | added | user76758 | @Andre: Going beyond being a "bad definition", is it even a definition if one hasn't already defined what "locally looks like" is supposed to mean (e.g., already have introduced a suitable 2-category in which these objects are meant to live, providing an a-priori notion of 1-morphism than makes the question posed somewhat moot)? Perhaps this is also implicit in your phrase "bad definition"... | |
| Dec 13, 2013 at 14:47 | comment | added | Bruno Martelli | As far as I know, the word "orbifold" was democratically invented during Thurston classes, and in his famous notes he introduces them as "a space locally modelled on Rn modulo finite group actions" (and then gives a formal definition realizing this idea). This term is also employed in algebraic geometry, but I don't know if the definitions agree. If possible, I would enjoy to have a definition of morphism which agrees with Thurston's simple (at least for me) definition. | |
| Dec 13, 2013 at 14:33 | history | answered | André Henriques | CC BY-SA 3.0 |