Timeline for Fundamental group of 3-manifold with boundary
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Aug 31, 2021 at 18:50 | history | edited | Autumn Kent | CC BY-SA 4.0 |
deleted 1 character in body
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| Apr 17, 2011 at 15:56 | vote | accept | Anton Petrunin | ||
| Mar 8, 2010 at 17:46 | comment | added | HJRW | Charlie, I don't think I understand your comment. Certainly, everything Richard has said is correct. Are you suggesting that something else isn't correct? | |
| Mar 8, 2010 at 0:39 | comment | added | Charlie Frohman | The answer by Richard Kent is correct. | |
| Dec 27, 2009 at 15:57 | comment | added | HJRW | Maharana, sure. D = M<sub>1</sub> U M<sub>2</sub>, where the M<sub>i</sub> are copies of M. Now the identifications M<sub>i</sub>->M agree on their boundaries, so extend to a map D->M. This is the retraction. | |
| Dec 27, 2009 at 7:05 | comment | added | Maharana | Could you explain a little bit how $M$ turns out to be a retract of its double $D$? | |
| Nov 21, 2009 at 17:51 | comment | added | HJRW | A less intelligent but more simple-minded algorithm is just to apply geometrization and deduce that the word problem is (uniformly) solvable in 3-manifold groups. If you know how to solve the word problem then it's easy to tell if a group is trivial. | |
| Nov 20, 2009 at 20:25 | comment | added | Autumn Kent | There is an algorithm to determine if $\pi_1(M)$ is trivial for compact $M$: If the boundary has a component that is not a sphere, then $M$ will have nontrivial homology. If the boundary is a union of spheres, then cap them off with balls to get a closed manifold $N$. Then, by the Poincare conjecture, which we now know, the question is whether or not $N$ is the three-sphere. You can use Rubinstein's algorithm to recognize the three-sphere to do this. | |
| Nov 20, 2009 at 18:15 | comment | added | Anton Petrunin | Do you know if there is an algorithm to decide if a group of 3-mnfld-with-bry is trivial? | |
| Nov 19, 2009 at 21:01 | vote | accept | Anton Petrunin | ||
| Apr 17, 2011 at 15:56 | |||||
| Nov 19, 2009 at 18:31 | history | edited | HJRW | CC BY-SA 2.5 |
Added reference to Scott's Theorem,
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| Nov 19, 2009 at 18:04 | history | answered | HJRW | CC BY-SA 2.5 |