Timeline for A notion of 2-dimensional tree
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Apr 8, 2022 at 12:15 | vote | accept | Agelos | ||
| Apr 8, 2022 at 12:15 | |||||
| Apr 3, 2022 at 17:39 | comment | added | Agelos | I wrote another update. | |
| Apr 3, 2022 at 17:05 | comment | added | Geva Yashfe | @SamNead No problem. Regarding the statement for higher dimensions: I made a conjecture about collapsibility for 2-trees (namely that they should collapse onto a one-dimensional subcomplex, in other words a graph). For $(n+1)$-trees, you could generalize my guess to the statement that they should collapse onto an $n$-dimensional CW complex. In particular if they are $n$-connected then they are contractible. In this sense $(n+1)$-manifolds with boundary are an example, but not a problematic one. | |
| Apr 3, 2022 at 13:39 | history | edited | Sam Nead | CC BY-SA 4.0 |
pointed out even worse problems.
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| Apr 3, 2022 at 13:30 | comment | added | Sam Nead | @GevaYashfe - Your technique applies to all manifolds with boundary. For example, properly embedded disks (that is, two-trees) separate points in three-manifolds. For example, in $S^2 \times I$. So all hope is lost. I'll add this to my answer. | |
| Apr 3, 2022 at 13:19 | comment | added | Sam Nead | @GevaYashfe - Fixed, and my apologies. | |
| Apr 3, 2022 at 13:18 | history | edited | Sam Nead | CC BY-SA 4.0 |
fix misspelt name.
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| Apr 3, 2022 at 9:52 | comment | added | Geva Yashfe | @SamNead By the way, please fix my name's spelling in your edit, if you can. Mathoverflow will not let me make the 1-character change. | |
| Apr 3, 2022 at 9:49 | comment | added | Geva Yashfe | @SamNead If my conjecture on collapsibility (in comments on the question) is correct, then the fundamental groups of these complexes are free, and simple-connectedness is a sufficient strengthening to imply contractibility. | |
| Apr 3, 2022 at 8:21 | history | edited | Sam Nead | CC BY-SA 4.0 |
Adde
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| Apr 3, 2022 at 8:15 | comment | added | Sam Nead | @GevaYashfe - Very nice counter-example. I’ll edit my post… as I now think that there is no obvious strengthening to give what the OP wants (other that CAT(0) cube complexes, I suppose?). | |
| Apr 2, 2022 at 22:51 | comment | added | Geva Yashfe | How about an annulus, e.g. $\{p \in \mathbb{R}^2 \mid 1\le \Vert p \Vert \le 2\}$? Any two points can be separated by a segment that has both endpoints on the same component of the boundary. In particular it is a 2-tree in the sense suggested above. But it is not simply connected. | |
| Apr 1, 2022 at 9:02 | comment | added | Sam Nead | That is a good question! I would hope that, with the correct set-up, these spaces are contractible… | |
| Apr 1, 2022 at 8:47 | comment | added | Agelos | Yes, I do like this variant! A nice feature is that 1-trees now coincide with standard trees. A proof of the Lemma would be very welcome. Do higher homotopy groups vanish as well? | |
| Apr 1, 2022 at 0:33 | history | edited | Sam Nead | CC BY-SA 4.0 |
added lemma
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| Apr 1, 2022 at 0:27 | history | edited | Sam Nead | CC BY-SA 4.0 |
wording
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| Apr 1, 2022 at 0:20 | history | answered | Sam Nead | CC BY-SA 4.0 |