Timeline for One-sided incompressible surface in 3-manifolds
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 24, 2021 at 17:31 | history | edited | Sam Nead | CC BY-SA 4.0 |
clarified
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| Aug 24, 2021 at 17:29 | comment | added | Sam Nead | @JoshHowie - Ok, this time I was not wrong, exactly... I have clarified. | |
| Aug 24, 2021 at 16:59 | comment | added | Josh Howie | Here $S$ is non-separating. After compressing as much as possible, some component will be incompressible and non-separating. That component cannot be 2-sided since $H_2(M;\mathbb{Z})=0$. | |
| Aug 24, 2021 at 16:57 | comment | added | Josh Howie | A compression can turn a 1-sided surface into a 2-sided surface, eg. think about the Klein bottle in $S^2\times S^1$. | |
| Aug 24, 2021 at 10:46 | comment | added | Sam Nead | @BrunoMartelli - Hempel writes out a careful definition of incompressible on page 58 of his book. I think of it as follows. Suppose that $(M, F)$ is a pair, with $M$ a closed three-manifold and $F$ a closed surface embedded in $M$. Then a "surgery disk" for $(M, F)$ is an embedding of pairs $(D, \partial D) \subset (M, F)$. If $\partial D$ is essential in $F$ then the surgery is "essential" and we have a "compression" etc. | |
| Aug 24, 2021 at 10:42 | comment | added | Sam Nead | @BrunoMartelli - ah, my brain skipped a beat. It is fixed now. | |
| Aug 24, 2021 at 10:40 | history | edited | Sam Nead | CC BY-SA 4.0 |
fixed bug pointed out by Bruno.
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| Aug 24, 2021 at 10:23 | comment | added | Zhiqiang | I see. The surface $S$ obtained in Ex.1 is nonorientable and 1-sided since $M$ is orientable. If $S=P^2$, then we are done by Ex.2. If $S$ is compressible, one can do surgery along the compressing disk to obtain a new surface $S'$ with $\chi(S') \ge \chi(S)+1$. Eventually one obtains an incompressible surface. | |
| Aug 24, 2021 at 9:17 | comment | added | Bruno Martelli | Non-orientable does not imply odd Euler characteristic, but you don't need that to deduce that it is 1-sided. On the other hand, I am a bit confused on the definition of incompressible surface in the 1-sided setting. | |
| Aug 24, 2021 at 8:55 | history | answered | Sam Nead | CC BY-SA 4.0 |