Timeline for Isomorphism of cobordisms
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Feb 14, 2012 at 17:10 | comment | added | knot | Also the cylindrical structure of the second cobordism is the identity, i.e., the isomorphism corresponding to $h$ above is the identity | |
| Feb 14, 2012 at 16:59 | comment | added | knot | Yes, by assumption we have a homeomorphism $\varphi: M \to \Sigma \times I$. The parametrization of this cobordism is given by $f_{\pm}$ on top and bottom respectively. What I want to have is a homeomorphism from this cobordism to a cobordism $(\Sigma \times I, \Sigma \times 0, \Sigma \times 1 )$, whose parametrization is given by the identity on the top and $f_{+}^{-1}hf_{-}$ on the botton as in the question above. Does it make sence? | |
| Feb 14, 2012 at 14:59 | comment | added | Kevin Walker | I don't understand your question (in the above comment). You wrote "Suppose $M$ is homeomorphic to a cylinder...", so the homeomorphism between $M$ and $\Sigma\times I$ exists by assumption; there is no need to construct it. | |
| Feb 14, 2012 at 2:02 | comment | added | knot | I think it should be trivial but I don't know how to prove it. Can you construct a homeomorphism? | |
| Feb 13, 2012 at 16:16 | history | edited | knot | CC BY-SA 3.0 |
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| Feb 13, 2012 at 13:42 | comment | added | Kevin Walker | I agree with what Ryan said. If it's just a question of whether the homeomorphism between the 3-manifolds commutes with the boundary parameterizations, then the answer to you question is clearly yes. Also, if anyone cares, this question is a continuation of mathoverflow.net/questions/87567 | |
| Feb 13, 2012 at 8:16 | history | edited | knot | CC BY-SA 3.0 |
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| Feb 13, 2012 at 8:07 | comment | added | Ryan Budney | I don't have Turaev's book in front of me but it looks like either the result follows immediately or it might be slightly sensitive to the definition of "parametrization" together with the structure of the mapping class group of $\Sigma$. It would help if you could make your question more self-contained, including the conventions Turaev uses. | |
| Feb 13, 2012 at 8:01 | history | asked | knot | CC BY-SA 3.0 |