Timeline for Is anything known about this braid group quotient?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Jan 12, 2023 at 17:33 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
http -> https (the question was bumped anyway)
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| Jan 12, 2023 at 17:01 | history | edited | Glorfindel | CC BY-SA 4.0 |
3 broken links fixed, cf. https://meta.mathoverflow.net/q/5301/70594
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| May 25, 2011 at 10:26 | comment | added | gowers | In a very elementary way I observed that when thinking about it. Using the third relation (the one that takes a string and passes it round all the other strings) and applying it to each string in turn, you get not a single twist but a double twist. | |
| May 25, 2011 at 4:01 | comment | added | Ryan Budney | The two are almost the same groups. The key relation is the fibre bundle $$Diff(S^2,n) \to Diff(S^2) \to C_n(S^2)/\Sigma_n$$ which means that you have a short exact sequence where the base group is $\pi_0 Diff(S^2,n)$, the middle group is the braid group of the sphere, and the fiber/kernel is $\mathbb Z_2$. So it's just a $\mathbb Z_2$-central extension between the two groups. | |
| May 25, 2011 at 3:51 | history | edited | Daniel Moskovich | CC BY-SA 3.0 |
error noted
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| May 25, 2011 at 3:47 | comment | added | Daniel Moskovich | @gowers: You're absolutely right. This answer is incorrect; and indeed, Mosher and Hamenstadt's results don't seem directly relevant to the spherical braid group as far as I can see. | |
| May 23, 2011 at 14:15 | comment | added | gowers | Looking again at the paper by Gillette and van Buskirk, I see that (if I understand correctly) my group is the mapping class group but not the spherical braid group: the mapping class group adds an extra relation that allows you to twist the entire bottom of the braid through a full turn. So it's really the mapping class group that I'm interested in, though obviously the two are closely related. | |
| May 23, 2011 at 10:01 | comment | added | gowers | Ian Agol says in a comment below that Mosher showed that mapping class groups are automatic and hence that there is a polynomial-time algorithm for the word problem. Am I missing something here, or does that mean that the problem you mention was solved in 1995 (the date of Mosher's paper)? | |
| May 19, 2011 at 16:02 | history | edited | Daniel Moskovich | CC BY-SA 3.0 |
added 256 characters in body
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| May 19, 2011 at 15:57 | comment | added | Daniel Moskovich | OK- so that's my ignorance. I'll edit. | |
| May 19, 2011 at 15:39 | comment | added | Andy Putman | @Daniel : The mapping class group literature (say, starting from Thurston's work in the late '70's) is enormous. A substantial portion of it is general enough to cover the case of the punctured sphere (ie the spherical braid group). | |
| May 19, 2011 at 15:14 | comment | added | Daniel Moskovich | This is the impression I got from looking at the literature- all the papers which seriously study these groups seem to be from the 1960's. Do you know of much between, say, 1970 and 2000? If so, I'll edit that in! | |
| May 19, 2011 at 15:09 | comment | added | Autumn Kent | I'm not sure why you keep saying these groups have been forgotten. The mapping class groups of punctures spheres have been studied pretty steadily. "Forgotten" because people don't call them braid groups? | |
| May 19, 2011 at 14:44 | history | edited | Daniel Moskovich | CC BY-SA 3.0 |
added 297 characters in body
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| May 18, 2011 at 21:08 | history | edited | Daniel Moskovich | CC BY-SA 3.0 |
fixed grammar
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| May 18, 2011 at 20:01 | history | answered | Daniel Moskovich | CC BY-SA 3.0 |