Timeline for If a polyhedron in $\mathbb{R}^3$ has local intersections, does it also have more global intersections?
Current License: CC BY-SA 4.0
8 events
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| Mar 23, 2023 at 13:54 | comment | added | Martin Tancer | PL means piecewise linear. (Not linear). This is one of the reasons why I say "As stated, the answer is no." If the author intends to ask a different question, then it should be rephrased. | |
| Mar 23, 2023 at 13:12 | comment | added | მამუკა ჯიბლაძე | Does not the question presume that PL in this case means linear on each simplex? Because if one interprets your example in this way, you will have intersecting images of disjoint simplices, no? | |
| Mar 23, 2023 at 11:01 | comment | added | Martin Tancer | I have added a simple 1-dimensional picture which hopefully clarifies the move. (I couldn't make it immediately when first posting the answer.) The main point is that PL maps allow subdivision of simplices. Then it is very easy to make self intersections and there are many ways how to get them. | |
| Mar 23, 2023 at 10:58 | history | edited | Martin Tancer | CC BY-SA 4.0 |
added 125 characters in body
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| Mar 23, 2023 at 8:12 | comment | added | მამუკა ჯიბლაძე | Should not the answer include this argument? Not that I do follow it well enough... | |
| Mar 23, 2023 at 8:08 | comment | added | Martin Tancer | PL is not a problem. One way to phrase this is to take a subdivision of $\sigma$ containing in the interior some other 2-simplex $\tau$ and a disjoint vertex $v$. Then the construction that I have described is essentially van Kampen's finger move from $\tau$ around $v$. These finger moves are PL. Everything can be performed locally in close neighborhood of $\sigma$ thus no additional intersections are introduced. | |
| Mar 23, 2023 at 7:53 | comment | added | მამუკა ჯიბლაძე | You have to remain PL. Is it obvious that it can be made an almost-embedding in the PL way? Only images of non-disjoint simplices are allowed to additionally intersect. | |
| Mar 23, 2023 at 7:38 | history | answered | Martin Tancer | CC BY-SA 4.0 |