Timeline for Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?
Current License: CC BY-SA 3.0
22 events
| when toggle format | what | by | license | comment | |
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| Jun 5 at 6:08 | answer | added | Pietro Majer | timeline score: 2 | |
| Jun 4 at 2:10 | comment | added | Nick Mendler | @ToddTrimble It has been shown over on What is the fewest number of points you must delete from R3 to make it not simply connected? that $\mathbb{R}^3\setminus C$ is simply connected as long as the cardinality of $C$ is less than that of the continuum $|C|<|2^{\mathbb{N}}|.$ When CH fails, this is a stronger result than what we get from the BCT. Also I posted a solution along the BCT line that doesnt use transversality and doesnt assume the ambient space to be smooth. | |
| Jun 3 at 16:43 | answer | added | Andy Putman | timeline score: 11 | |
| Jun 2 at 2:59 | answer | added | Nick Mendler | timeline score: 5 | |
| Jun 9, 2017 at 23:20 | review | Close votes | |||
| Jun 10, 2017 at 0:06 | |||||
| Sep 6, 2015 at 3:03 | review | Close votes | |||
| Sep 6, 2015 at 8:35 | |||||
| Aug 31, 2015 at 10:19 | comment | added | Neil Strickland | Adams mentions in "Lectures on Lie Groups" that there is a good theory of transversality and homotopy formulated using Hausdorff dimension, but he does not give a reference and I have never seen one. Assuming that that is correct, it would immediately answer the question asked here. | |
| Aug 30, 2015 at 22:02 | comment | added | Martin M. W. | Is it obvious that transversality arguments work for weird non-differentiable curves? In a manifold you know any curve is homotopic to a differentiable one, but showing that fact in $\mathbb{R}^3 \setminus \mathbb{Q}^3$ doesn't seem any easier than the original question. | |
| Aug 30, 2015 at 21:14 | comment | added | Todd Trimble | @MikeMiller Why not write up the argument carefully as an answer before the question is closed? The comment as written is a little too telegraphic for me to follow easily, and I don't know which thing of Hirsch you mean. A careful, detailed, self-contained answer would be peachy. | |
| Aug 30, 2015 at 21:03 | comment | added | mme | @ToddTrimble: This is more general than just a countable collection of points; there is a version of the transversality theorem for maps $g: M \to N$ transverse to $f(A) \subset N$ noncompact (and not necessarily closed): transverse maps are still dense, just not necessarily open, in the space of all maps - even when you specify $\partial g$ ahead of time. As usual, our manifolds here are second countable, or this is completely false. This is somewhere in Hirsch. | |
| Aug 30, 2015 at 20:28 | comment | added | user44143 | For the title question, maybe some examples will help: Consider the circle $x^2+y^2=\sqrt[3]{4}$ or the square $\max(|x|,|y|)=\sqrt{2}$ in $R^2-Q^2$. Every contraction of them to a point in $R^2$ will go through all of their interior points in $R^2$, and in particular will not avoid $Q^2$. But the corresponding curves on the $z=0$ plane of $R^3-Q^3$ are both homotopic to curves on the $z=\sqrt{2}$ plane of $R^3-Q^3$, where they can be contracted. | |
| Aug 30, 2015 at 18:53 | comment | added | Ryan Budney | It's essentially the same transversality argument one uses to show $\pi_k S^n$ is trivial for $k < n$. Only you notice you can make the map simultaneously transverse to a countable collection of manifolds. | |
| S Aug 30, 2015 at 17:21 | history | suggested | Stefan Hamcke | CC BY-SA 3.0 |
added tags
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| Aug 30, 2015 at 16:58 | comment | added | Todd Trimble | While I harbor some doubts that one could do better than Martin's answer below, why not explain this alternative argument? It could be educational, even for professionals. | |
| Aug 30, 2015 at 16:42 | review | Suggested edits | |||
| S Aug 30, 2015 at 17:21 | |||||
| Aug 30, 2015 at 6:22 | comment | added | Ryan Budney | There's a direct argument for this using the transversality-extension theorem. | |
| Aug 30, 2015 at 5:55 | vote | accept | Nick R | ||
| Aug 29, 2015 at 23:08 | review | Close votes | |||
| Aug 30, 2015 at 9:08 | |||||
| Aug 29, 2015 at 19:55 | answer | added | Martin M. W. | timeline score: 104 | |
| Aug 29, 2015 at 18:59 | history | edited | Nick R | CC BY-SA 3.0 |
added 4 characters in body; edited title
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| Aug 29, 2015 at 18:51 | review | First posts | |||
| Aug 29, 2015 at 19:02 | |||||
| Aug 29, 2015 at 18:46 | history | asked | Nick R | CC BY-SA 3.0 |