Timeline for Sperner's lemma and paths from one side to the opposite one in a grid
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| May 24, 2020 at 14:40 | comment | added | Claus | I just wanted to share with you, in case you are interested in this: a nice proof from user Oliver Clarke that there is more than 1 crossing path (along diagonals in the grid) math.stackexchange.com/q/3689297/782412 | |
| May 18, 2020 at 19:27 | history | edited | Wlod AA | CC BY-SA 4.0 |
Precision!!! (great :) ).
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| May 18, 2020 at 16:45 | comment | added | Claus | Let us continue this discussion in chat. | |
| May 18, 2020 at 16:45 | comment | added | Claus | I studied your proof in more detail now. I think it goes to the heart of the matter. I also like my proof with Sperner Lemma, but your proof has the advantage that it clarifies the situation better and explains better "why it is true". Thanks again for developing it! | |
| May 18, 2020 at 12:23 | comment | added | Wlod AA | @Claus, "whether you feel the crossing-along-diagonals lemma could prove the No Retraction Theorem?" #### Oh, yes, from the first moment (I should write it down), as many others would do. Hex and other games were mentioned in this thread, the intuition is clear (at one moment, I made a wrong argument, then my view temporary has switched for a wrong reason). ### With XX century patience, you may enjoy a lot of classic "Dimension Theory" by Witold Hurewicz and Henry Wallman. And more up to date are texts by Ryszard Engelking. | |
| May 18, 2020 at 8:05 | comment | added | Claus | thanks again. Especially your second comment, I was not aware of this. I realize I should have made my second comment clearer. What I intended to say, I tried to use this crossing-along-diagonals lemma to prove Brouwer FPT, but I failed. Therefore I was interested in your opinion, whether you feel the crossing-along-diagonals lemma could prove the No Retraction Theorem? | |
| May 18, 2020 at 7:18 | comment | added | Wlod AA | The opposite implication is even easier. If you had a said retraction then compose it with the antipodal map of $\ S,\ $ and you get a map of the ball into itself free of any fixed points (the image would even reside in $\ S$). | |
| May 18, 2020 at 7:10 | comment | added | Wlod AA | @Claus, a classical argument: let continuous $\ f:B\to B\ $ was such that $\ \forall_{x\in B} f(x)\ne x,\ $ where $\ B\ $ is the unit Euclidean (rounded) n-ball. Then $\ r:B\to S\ $ (where $\ S\ $ is the boundary sphere) such that from $\ f(x)\ $ you shoot at $\ r(x)\ $ through $\ x\in B\ $ is a continuous retraction of $\ B\ $ onto $S$. Thus the non-retraction theorem implies the fpp theorem. | |
| May 18, 2020 at 5:18 | comment | added | Claus | my third comment for further research I am planning to undertake is about higher dimensions: please see here the proof (just a summary) of the crossing-along-diagonals lemma using the Sperner lemma math.stackexchange.com/a/3677664/782412. The interesting thing is that the Sperner Lemma also holds for higher dimensions. So my thinking is that the crossing-along-diagonals lemma could actually also be true for higher dimensions | |
| May 18, 2020 at 5:13 | comment | added | Claus | please let me also know if you think the crossing-along-diagonals lemma can prove the No Retraction Theorem. As I mentioned earlier, I failed to use it to prove Brouwer's FPT. | |
| May 18, 2020 at 5:09 | comment | added | Claus | thanks a thousand times for this proof. I think your idea to relate it to the classical topological dimension theorem is very powerful. Please let me know your opinion, do you think this crossing-along-diagonals lemma (puzzle) could also be used to prove the topological dimension theorem? This would create a fascinating equivalence of theorems. | |
| May 17, 2020 at 22:54 | history | edited | Wlod AA | CC BY-SA 4.0 |
cosmetics
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| May 17, 2020 at 22:46 | history | answered | Wlod AA | CC BY-SA 4.0 |