Timeline for what is the meaning of a curve $C$ representing Identity in fundamental group?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Nov 28, 2011 at 14:31 | comment | added | Autumn Kent | I am saying that if being interior to a ball were the same as being trivial in the fundamental group, then you would have a new proof of the Poincare conjecture, which suggests that the assumption is false. The fact that they are not the same is easy to see: one component of the Whitehead link is null homotopic in the exterior of the other, but is not interior to a ball there. | |
| Nov 28, 2011 at 5:19 | comment | added | yanqing | @binyu : Are you from tongji university, Shanghai? | |
| Nov 27, 2011 at 21:47 | comment | added | Autumn Kent | Also, you need more reputation to comment. This is to prevent you from commenting until you've been around a while. | |
| Nov 27, 2011 at 21:45 | comment | added | Autumn Kent | Being interior to a ball is not the same as being trivial in the fundamental group. In fact, it is a theorem of Bing that M is the 3-sphere if and only if every knot is interior to a ball. So, the Poincare conjecture would follow from your "visual meaning." | |
| Nov 27, 2011 at 21:24 | comment | added | Bin Yu | Where do you use $[K]=0$ in $\pi_1{M}$? I feel that your answer is an explain of Dehn's lemma. I guess that "$[K]=0$" visual meaning is "$K\subset D^3 \subset M$" where $D^3$ is a 3 ball(similar to knot in $S^3$). | |
| Nov 26, 2011 at 12:05 | history | edited | Sam Nead | CC BY-SA 3.0 |
Its now fairly detailed!
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| Nov 26, 2011 at 12:01 | comment | added | Sam Nead | @Yanqing - My answer gives a necessary and sufficient condition for $K$ to bound an embedded disk in $M$. I'll restate this at the end of the answer. | |
| Nov 26, 2011 at 1:17 | comment | added | yanqing | To Sam: So it is not clear whether $K$ bounds an embedded or not. | |
| Nov 26, 2011 at 1:12 | vote | accept | yanqing | ||
| Nov 25, 2011 at 21:58 | history | edited | Sam Nead | CC BY-SA 3.0 |
many typos!
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| Nov 25, 2011 at 17:35 | history | answered | Sam Nead | CC BY-SA 3.0 |