Timeline for Word problem for fundamental group of submanifolds of the 4-sphere
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 18, 2010 at 4:45 | vote | accept | Ryan Budney | ||
| Jul 30, 2010 at 1:23 | comment | added | Ian Agol | @ Greg: I think you're right - I was worried about a neighborhood of a branch point (like a cone on a knot) not having a tubular neighborhood (tubular neighborhoods have obvious retracts to the complex, guaranteeing $\pi_1$-injectivity). But I think one can still take a neighborhood in which the complex will be $\pi_1$-injective using Van Kampen. | |
| Jul 23, 2010 at 17:23 | history | edited | Ryan Budney | CC BY-SA 2.5 |
update - question answered
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| Jul 21, 2010 at 16:04 | comment | added | Greg Kuperberg | @Agol: If it's a PL embedding, then what's the problem? It may not have locally flat 2-cells, but it has a refinement with locally flat 2-cells. Isn't that good enough for the question? | |
| Jul 21, 2010 at 15:50 | comment | added | Ian Agol | You can embed a finite 1-vertex complex in $\mathbb{R}^4$. See: mathoverflow.net/questions/30238/… However, I'm not sure that the complex has a regular neighborhood - some of the 2-cells might have branched points. So it probably won't be $\pi_1$-injective into an open submanifold. | |
| Jul 21, 2010 at 14:29 | answer | added | Autumn Kent | timeline score: 15 | |
| Jul 21, 2010 at 14:04 | comment | added | Autumn Kent | Do you want the group to be finitely presented? | |
| Jul 21, 2010 at 12:19 | comment | added | Jon Bannon | What if one asks the same question requiring only that $\pi_{1}(M)$ fails to be residually finite? | |
| Jul 21, 2010 at 11:17 | history | asked | Ryan Budney | CC BY-SA 2.5 |