Timeline for Connected sum of surfaces
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| S Apr 24, 2017 at 14:37 | history | suggested | Dario | CC BY-SA 3.0 |
fixed latex
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| Apr 24, 2017 at 14:13 | review | Suggested edits | |||
| S Apr 24, 2017 at 14:37 | |||||
| Feb 11, 2012 at 23:23 | comment | added | William | It seems like it comes down to the case where you are embedding the disk into an oriented surface with embeddings who have different orientation parity (reversing/preserving), because in every other case you get an isotopy of the embeddings. Maybe there is an elementary cut-and-paste argument you could give for this case. After all, being able to prove with cut-and-paste techniques is what makes the theory of surfaces so elementary. | |
| Feb 11, 2012 at 12:19 | comment | added | Baptiste Calmès | @William. I am talking about surfaces. I know about the classical counter example in dimension 4 that Johannes Ebert was mentioning (captured by homology, by the way). However, it has to be true for surfaces, even non-compact ones, in view of the classification result mentioned by Agol. But this is definitely not a simple argument, and given no one has come up with an obvious trick, it certainly isn't intuitive before classification that connected sums of oriented surfaces are homeomorphic, whatever glueing you use. | |
| Feb 9, 2012 at 22:29 | comment | added | William | Ah, well then in that case it is as Johanns said: there's no guarantee that the connected sums are even homotopy equivalent without some restriction on the embeddings or on the manifolds. | |
| Feb 9, 2012 at 9:33 | comment | added | Baptiste Calmès | Thanks for your answer. In my mind, though, the important case is the one where the maps are not isotopic (i.e. the surface is orientable, and the disks are glued in opposite ways). | |
| Feb 9, 2012 at 0:32 | comment | added | William | Whoops, I may have made a huge gaff: does Isotopy Extension Property apply to Topological manifolds? I know the original question is just about surfaces, but I'm wondering more generally. | |
| Feb 9, 2012 at 0:22 | history | answered | William | CC BY-SA 3.0 |