Timeline for Wild spheres in higher dimensions
Current License: CC BY-SA 4.0
22 events
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| Dec 30, 2021 at 19:47 | vote | accept | amd1234 | ||
| Dec 23, 2021 at 19:48 | answer | added | Shijie Gu | timeline score: 8 | |
| Dec 21, 2021 at 20:56 | comment | added | Shijie Gu | The full power of engulfing techniques requires the limitation n>4 because then there is room to control the top dimension of the space. Probably that will explain the distinction. The 4d case is sort of outcasted. But it’s handled by Kirby. See the union of flat (n-1)-balls is flat inR^n, Bull Amer Math Soc 74 (1968) 614-617 | |
| Dec 21, 2021 at 20:53 | comment | added | Shijie Gu | Historically the differences between the 3D and >3D cases were first spotted by Cantrell’s dissertation, where he proved that a sphere which is locally flat modulo a point is flat. But his proof is a little too technical. A widely used technique is engulfing. Heuristically one can expand an open subset of a manifold to engulf a subpolyhedron P, provided certain dimension, connectivity and finiteness conditions are satisfied. Among those the codimension P is at least 3. The full power is engulfing techniques require the limitation n>4 because then there is room to control the top dimension. | |
| Dec 21, 2021 at 20:25 | comment | added | Danny Ruberman | You could consult Kirby's paper, On the set of non-locally flat points of a submanifold of codimension one. Ann. of Math. (2) 88 (1968), 281–290. It's available on his web page: math.berkeley.edu/~kirby. The paper doesn't discuss the reason for the assumption $\dim > 3$. | |
| Dec 21, 2021 at 19:47 | comment | added | amd1234 | @IgorBelegradek PS I did not link the wiki page someone edited over me (I removed it now) | |
| Dec 21, 2021 at 19:39 | history | edited | amd1234 | CC BY-SA 4.0 |
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| Dec 21, 2021 at 19:38 | comment | added | Igor Belegradek | One place where $n>3$ is used in the proof is exercise 2.5.2 that in $\mathbb R^{n>3}$ any arc that is a countable union of points and line segments is flat (i.e. can be unknotted). | |
| Dec 21, 2021 at 19:36 | review | Close votes | |||
| Dec 26, 2021 at 3:04 | |||||
| Dec 21, 2021 at 19:30 | comment | added | amd1234 | @RyanBudney Theorem 2.9.3 states: if $\Sigma\subset S^n$, $n>3$ is an embedded codimension 1 sphere and $\Sigma$ has a bicollar that is PL modulo the preimage of p, then $\Sigma$ is flat. | |
| Dec 21, 2021 at 19:20 | comment | added | amd1234 | Please correct me if I say something dubious as I am new to this subject. As far as I understand locally flat is equivalent to existence of a local bicollar. When I write 'wild in one point' I mean that it is locally flat everywhere but a point. By lemma 2.9.2 in DV we get the existence of a bicollar pinched at the wild point that is in the statement of Theorem 2.9.3. As for sphere of codimension higher than one I am not aware of any results (apart from Cantarelli-Edwards on $S^1$) so if someone knows anything on the matter I would be interested. | |
| Dec 21, 2021 at 19:19 | comment | added | Ryan Budney | I don't have access to Daverman-Venema, could the poster supply the statement of the theorem that "forbids wild spheres"? | |
| Dec 21, 2021 at 19:15 | history | edited | amd1234 | CC BY-SA 4.0 |
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| Dec 21, 2021 at 17:30 | comment | added | Igor Belegradek | I gather you are asking why Theorem 2.9.3 in Daverman-Venema "Embeddings in Manifolds" works in dimensions $>3$ but not in dimension $3$. Its statement seems more technical than what you claim. Why is it the same as what you claim? | |
| Dec 21, 2021 at 16:17 | history | edited | amd1234 | CC BY-SA 4.0 |
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| Dec 21, 2021 at 16:13 | comment | added | amd1234 | see Daverman-Venema "Embeddings in Manifolds" section 2.9 for the case of one point | |
| Dec 21, 2021 at 15:27 | comment | added | Igor Belegradek | I am sorry, where is it shown that "there are no wild spheres in higher dimensions $n>3$ that are wild at one point"? I cannot find the statement in the the wikipedia page you link, and I doubt it is true. | |
| Dec 21, 2021 at 15:10 | history | edited | Joe Silverman | CC BY-SA 4.0 |
added wikipedia link for definition of wild sphere
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| Dec 21, 2021 at 12:48 | history | edited | YCor | CC BY-SA 4.0 |
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| Dec 21, 2021 at 12:45 | history | edited | amd1234 | CC BY-SA 4.0 |
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| S Dec 21, 2021 at 12:38 | review | First questions | |||
| Dec 21, 2021 at 13:25 | |||||
| S Dec 21, 2021 at 12:38 | history | asked | amd1234 | CC BY-SA 4.0 |