Timeline for Isotopy class of closed 2-ball embedded in R^3
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Feb 17, 2016 at 16:42 | vote | accept | Jaeba | ||
| Feb 12, 2016 at 17:12 | answer | added | Sylvain Maillot | timeline score: 3 | |
| Feb 12, 2016 at 15:35 | comment | added | Sylvain Maillot | There exist so-called wild disks in $R^3$ (cf. J. R. Stallings, Uncountably many wild disks, Ann. of Math. (2) 71 (1960), 185— 186.) Now the definition of a wild disk is a disk which is not equivalent to a PL disk, and the definition of "equivalent" is the following : $D_1$ is equivalent to $D_2$ if there is a self-homeomorphism of $R^3$ which takes $D_1$ to $D_2$. However, I don't see any reason why two inequivalent disks couldn't be isotopic. | |
| Feb 12, 2016 at 9:07 | comment | added | YCor | It's not clear to me that an embedding mapping the sphere to a horned sphere can't be isotopic (= homotopic where each step is an embedding) to a standard embedding. Isn't the horned sphere defined as a uniform limit of smooth embeddings? | |
| Feb 12, 2016 at 6:48 | comment | added | Ryan Budney | I'm not sure your question is simple. You are talking about topological embeddings, not smooth embeddings. Your idea about the disc being contractible, if you follow it through, is a smooth argument. I think your question is complicated because you have overlooked how difficult it is to construct all possible topological embeddings. | |
| Feb 12, 2016 at 5:51 | history | edited | Jaeba | CC BY-SA 3.0 |
added 256 characters in body
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| Feb 12, 2016 at 5:25 | history | edited | Jaeba | CC BY-SA 3.0 |
added more allowed assumptions
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| Feb 12, 2016 at 4:39 | comment | added | Jaeba | Very good observation! Simple topological questions always seem to result in some pathological counterexample that redefines our intuition. Since the complement of the usual disk is simply connected while the complement of the Alexander Horned Sphere (AHS) is not, the next question would be, is the homotopy type of the complement of an embedding an isotopy invariant? It appears so. This only gets more and more confusing/exciting. | |
| Feb 12, 2016 at 3:46 | comment | added | ARupinski | If one of your embeddings consists of an Alexander Horned Sphere with an open disk removed then it seems that should not be isotopic to the embedding of a disk as a filled unit circle in some plane. Perhaps someone with more of a topology background can confirm if I this is indeed true, or conversely correct me if my intuition about puncturing a Horned Sphere is in fact wrong. | |
| Feb 12, 2016 at 3:22 | review | First posts | |||
| Feb 12, 2016 at 4:01 | |||||
| Feb 12, 2016 at 3:21 | history | asked | Jaeba | CC BY-SA 3.0 |