Timeline for Can the infinite jungle gym surface be expressed by an exhaustion of compact surfaces with one boundary component?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Mar 23, 2021 at 18:37 | comment | added | Moishe Kohan | Cross-posted here. In general, you should not post a questions simultaneously here and at MSE. | |
| Mar 21, 2021 at 5:57 | comment | added | Kevin Casto | Consider an exhaustion of this surface by an $n \times n \times n$ "cube" (for each $n$), with $6(n+1)^2$ boundary circles coming from all the "rods" sticking out. Now draw a giant circuituous circle (simple closed curve) on the surface that encloses all these boundary circles. | |
| Mar 21, 2021 at 4:15 | comment | added | Roberta | (note: I was not the person who downvoted, and in fact haven't registered my account and thus do not have voting rights at all!) | |
| Mar 21, 2021 at 3:51 | comment | added | Roberta | The answer to this question is "yes". This is an example of an "infinite Loch Ness surface"; see arxiv.org/pdf/1701.07151.pdf for a discussion of these surfaces and references to the classification of surfaces of infinite type. All infinite-genus $1$ ended surfaces are diffeomorphic, and thus are diffeomorphic to the canonical example of the union of the genus-$g$ surfaces $\Sigma_{g,1}$ with one boundary component as $g$ goes to infinity (the boundary curves here give you the curves you are looking for). I'm not a good artist, so I'll leave others to give you explicit pictures. | |
| Mar 21, 2021 at 3:47 | history | edited | Fernando Oliveira | CC BY-SA 4.0 |
added 63 characters in body; edited title
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| Mar 21, 2021 at 1:43 | history | asked | Fernando Oliveira | CC BY-SA 4.0 |