Timeline for Integer surgery on $S^3$
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| May 8, 2020 at 12:09 | history | edited | Steve | CC BY-SA 4.0 |
added 1 character in body
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| Apr 9, 2020 at 12:19 | vote | accept | Steve | ||
| Apr 9, 2020 at 12:14 | vote | accept | Steve | ||
| Apr 9, 2020 at 12:19 | |||||
| Apr 9, 2020 at 11:12 | answer | added | Mehdi Yazdi | timeline score: 3 | |
| Apr 9, 2020 at 5:58 | comment | added | Ryan Budney | Perhaps it helps to think about a simpler example. Glue the boundary of one interval $[0,1]$ to the boundary of another interval $[0,1]$. There are two gluing homeomorphisms of the boundary $\partial [0,1] = \{0,1\}$. But the resulting manifold, either way, is homeomorphic to the circle. They are not the same manifolds set-theoretically, but up to an essentially canonical homeomorphism, they are the same. This is because the homeomorphisms of the boundary extend over the interval. | |
| Apr 9, 2020 at 5:56 | comment | added | Ryan Budney | I did answer your question: the homeomorphism is not determined by the integer. But the resulting manifold is determined by the integer, up to an essentially canonical homeomorphism. | |
| Apr 9, 2020 at 3:44 | comment | added | Steve | Thank you but that still does not answer my question though, When specify the surgery, one declares the meridian of the first torus goes to the framing of the knot on the second torus and that is sufficient to completely define this operation ( for me it seem that what is needed is to define the entire map between the two tori which is an element of $GL_2\mathbf{Z}$ as you mentioned ), my question is why just the mapping between these two curves this is sufficient (again given what you said that a homeomorphism of a torus is determined by an element of $GL_2\mathbf{Z}$ )? | |
| Apr 9, 2020 at 3:33 | comment | added | Ryan Budney | The homeomorphism is not determined by the integer. Homeomorphisms of a torus up to isotopy are determined by an element of $GL_2 \mathbb Z$. But the resulting manifold is determined by the integer -- this is because the resulting gluing manifolds will be homeomorphic. A solid torus has non-trivial homeomorphisms, up to isotopy. In particular if you fix a meridian of a solid torus, and demand the homeomorphism be orientation preserving, all such homeomorphisms are determined by the linking number of the longitude. | |
| Apr 9, 2020 at 3:25 | history | asked | Steve | CC BY-SA 4.0 |