Timeline for Surface in a product domain
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 11, 2021 at 3:29 | vote | accept | Adterram | ||
| Jun 11, 2021 at 3:05 | comment | added | Sam Nead | By the way, if I’ve answered your original question, it is polite (and tidy) to accept the answer. :) | |
| Jun 11, 2021 at 2:32 | comment | added | Sam Nead | it meets the internal one-cell an odd number of times. We now perform isotopies of $N$ to ensure that all intersections of $N$ with the "vertical" rectangles (two-cells in the interior of $M \times [0, 1]$) are horizontal arcs. Also, we can arrange that intersections of $N$ with the three-cell are disks. We deduce that $N$ meets the internal one-cell exactly once. A final isotopy gives the result. Here is a more algebraic proof: compress $N$ to be incompressible, appeal to Dehn's lemma to deduce that $N$ is essential, and then use the classification of subgroups of surface groups. | |
| Jun 11, 2021 at 2:26 | comment | added | Sam Nead | I poked around, and could not find an obvious place to point to on-line. I'll guess that this appears in either Jaco's book or Hempel's - unfortunately I don't have them to hand. The proof is not so difficult, however. Suppose that $M$ has genus $g$. Fix a cell structure on $M$ with one vertex, $2g$ edges, and one $4g$-gon (as the two-cell). Cross this with the closed interval to get a cell structure on $M \times [0, 1]$. Isotope $N$ to be in general position with respect to the one- and two-cells of the cell structure on $M \times [0, 1]$. Since $N$ separates the boundary components, | |
| Jun 11, 2021 at 1:14 | comment | added | Adterram | Could you give a reference for this fact? | |
| Jun 10, 2021 at 19:49 | comment | added | Sam Nead | Yes. If $N$ separates the boundary components, and is homeomorphic to $M$, then there is an ambient isotopy taking $N$ to $M$. | |
| Jun 10, 2021 at 8:36 | comment | added | Adterram | Is the original conclusion true if we assume $N$ is homeomorphic to $M$? | |
| Jun 9, 2021 at 13:55 | history | edited | Sam Nead | CC BY-SA 4.0 |
Removed snark.
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| Jun 9, 2021 at 12:45 | history | answered | Sam Nead | CC BY-SA 4.0 |