Timeline for Compelling evidence that two basepoints are better than one
Current License: CC BY-SA 2.5
24 events
| when toggle format | what | by | license | comment | |
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| Aug 8, 2017 at 6:02 | answer | added | Daniel Moskovich | timeline score: 4 | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Dec 8, 2014 at 16:44 | comment | added | Ryan Budney | @Ronnie, I was suggesting the map from the space $X$ to its quotient, modulo the action. | |
| Dec 8, 2014 at 11:06 | comment | added | Ronnie Brown | @Ryan: You comment that "you have a covering space, so study that"; but which one? A universal cover of $X$ is defined by $X$ and a base point $x$. It is in fact the star, or costar, of $\pi_1 X$ at a base point $x$. See my preprint page for a recent talk at Galway which starts with 5 anomalies in algebraic topology, most of which are resolved using groupoids and/or cubes in some way. | |
| Apr 21, 2013 at 18:52 | answer | added | Peter May | timeline score: 20 | |
| Apr 21, 2013 at 16:40 | answer | added | John Klein | timeline score: 9 | |
| Apr 21, 2013 at 14:31 | answer | added | Lee Mosher | timeline score: 13 | |
| Nov 15, 2011 at 3:45 | vote | accept | Daniel Moskovich | ||
| Nov 21, 2010 at 10:32 | answer | added | Ronnie Brown | timeline score: 50 | |
| Oct 8, 2010 at 15:48 | answer | added | Bruno Martelli | timeline score: 7 | |
| Oct 4, 2010 at 11:56 | answer | added | Laurence Taylor | timeline score: 7 | |
| Oct 4, 2010 at 9:24 | answer | added | Greg Graviton | timeline score: 6 | |
| Oct 4, 2010 at 9:22 | answer | added | James Griffin | timeline score: 18 | |
| Oct 4, 2010 at 8:51 | comment | added | David Corfield | For some Grothendieckian support for Ronnie see the comment at bangor.ac.uk/~mas010/pstacks.htm which begins "...people are accustomed to work with fundamental groups and generators and relations for these and stick to it, even in contexts when this is wholly inadequate, namely when you get a clear description by generators and relations only when working simultaneously with a whole bunch of base-points chosen with care-or equivalently working in the algebraic context of groupoids, rather than groups." | |
| Oct 3, 2010 at 22:12 | comment | added | Dustin Clausen | Ryan, that sounds good -- thanks for the perspective. Here's a link: youtube.com/watch?v=kjX2MKaZ-rg | |
| Oct 3, 2010 at 22:04 | answer | added | Jeffrey Giansiracusa | timeline score: 37 | |
| Oct 3, 2010 at 21:53 | comment | added | Ryan Budney | @Dustin, a standard strategy here would be (1) if the action has a fixed point, use the fixed point and (2) if there is no fixed point, you have a covering space, so study that. | |
| Oct 3, 2010 at 21:44 | comment | added | Dustin Clausen | I'm sorry that I don't know much about geometric topology and can't answer the question, but let me point out something that you may already know: multiple basepoints are useful in the presence of symmetries. For example, let's say you have a connected space X with a Z/2 symmetry. Then H_1(X;Z) gets a Z/2-action; shouldn't this come from an action of Z/2 on \pi_1(X) via the isomorphism \pi_1^{ab}=H_1? If X has a fixed point we can base ourselves there and be fine, but otherwise it's nice to simultaneously use two conjugate basepoints to see the Z/2-action on \pi_1. | |
| Oct 3, 2010 at 19:32 | answer | added | André Henriques | timeline score: 30 | |
| Oct 3, 2010 at 19:22 | answer | added | Robert Bell | timeline score: 5 | |
| Oct 3, 2010 at 18:51 | comment | added | Daniel Moskovich | @Ryan- that's my worst fear (in the context of this question), and I hope it's not true. That the "gain" in avoiding basepoint-arcs and such nonsense is "always" offset by the cost of using groupoids; that there's no more to the story than that; and that the fundamental groupoid will never be of much use for the geometric topologist. | |
| Oct 3, 2010 at 18:47 | comment | added | Ryan Budney | Natural generalizations of braids seem to be things like tangles and string links, which aren't groupoids but they are categories. In that sense they fit into a natural framework of things like cobordism categories. I suppose I don't see the fundamental groupoid as much of a simplification -- the Seifert-Van Kampen theorem is essentially just as complicated in the groupoid setting, the only thing you get to avoid is putting in all the basepoint arcs after subdivision. This "gain" comes at the cost of using groupoids. How much you value the approach depends on what you need the tool for. | |
| Oct 3, 2010 at 18:31 | history | edited | Daniel Moskovich | CC BY-SA 2.5 |
added 9 characters in body
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| Oct 3, 2010 at 18:26 | history | asked | Daniel Moskovich | CC BY-SA 2.5 |