Timeline for Is every tree a deformation retract of the disk?
Current License: CC BY-SA 4.0
15 events
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| Nov 22, 2022 at 14:25 | comment | added | Isky Mathews | @PierrePC: This is a good point! The thing to focus on is not if an infinite tree itself is compact but whether we can embed it as something compact. Also, your description is something I had also seen in my mind when asking the question but again was not sure how to prove it. I'll think about the fatgraph construction mentioned above. | |
| Nov 22, 2022 at 13:27 | comment | added | Pierre PC | For similar reasons, the tree needs to be locally finite (each vertex has finite degree): roughly, for $x$ a vertex with infinite degree, one can choose $p_n$ with $|p_n-x|=1/n$ and different $p_n$ on different edges; the set $\{p_n\}_{n\geq1}$ is then closed in the graph but not in the disc. From this, I feel like there is a natural embedding of any locally finite tree in the disc, placing a root at the center, and its $\mathrm{grand}^n\mathrm{children}$ on the circle of radius $1-2^{-n}$ in such a way that the straight-line edges do not intersect. I would guess this is a deformation retract. | |
| Nov 22, 2022 at 12:56 | comment | added | Pierre PC | Just a remark: any graph embedding in the disk (open or closed, or any manifold for that matter) has to be at most countable. Indeed the set of vertices is discrete in the graph topology so it must be discrete, hence at least countable, in the subset topology as well. | |
| Nov 22, 2022 at 12:46 | comment | added | Isky Mathews | @HJRW: Yeah indeed! I was in a lecture during the asking of this question and while I was away typing things up I realised this quickly and came back to mention it but I guess it's very obvious, as you say. | |
| Nov 22, 2022 at 12:35 | comment | added | Jonny Evans | Are you allowing the "long line" as a tree? | |
| Nov 22, 2022 at 12:18 | comment | added | HJRW | It's easy to see that no infinite graph is compact: just consider the open covering by balls of radius $3/4$ centred on the vertices. | |
| Nov 22, 2022 at 12:12 | comment | added | Isky Mathews | @Wojowu: Just edited based on your comments. Nice remark about infinite trees! In fact, your remark about trees has made me wonder if there are any infinite, compact trees? | |
| Nov 22, 2022 at 12:11 | history | edited | Isky Mathews | CC BY-SA 4.0 |
Made question better defined and clarified the closed disk.
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| Nov 22, 2022 at 12:08 | comment | added | HJRW | (cont'd from above.)... Finally, if $T$ is a finite tree then $S(T)$ is a compact, connected surface of Euler characteristic 1, so is homeomorphic to a disc by the classification of surfaces. | |
| Nov 22, 2022 at 12:08 | comment | added | Wojowu | Also, are we talking about open or closed disk? In the context of fixed point results I would imagine the latter, but then every restract has to be compact, and not all (infinite) trees are compact. | |
| Nov 22, 2022 at 12:08 | comment | added | Isky Mathews | @Wojowu Yeah sorry that is the better question. I tried to get around this in my answer by saying "implicitly we're embedding the tree in $\mathbb{R}^2$ already" but you're right, probably that's something I should actually be looking for in the question. | |
| Nov 22, 2022 at 12:07 | comment | added | HJRW | One easy way to do this (for finite trees) is to give your tree the structure of a fatgraph. To do this, you put a cyclic ordering on the edges incident at each vertex, and there's also an obvious orientation condition on each edge. It's easy to prove by induction that any finite graph has the structure of a fatgraph. It's also easy to see that a fatgraph structure allows you to "thicken" your graph $G$ to a compact surface with boundary $S(G)$, in such a way that $S(G)$ deformation retracts to $G$... | |
| Nov 22, 2022 at 11:52 | comment | added | Wojowu | To talk about tree being a retraction of a disk you need that the tree be a subspace of the disk. Perhaps you mean to ask whether there is an embedding of the tree into the disk whose image is a (deformation) retract? | |
| S Nov 22, 2022 at 11:30 | review | First questions | |||
| Nov 22, 2022 at 15:32 | |||||
| S Nov 22, 2022 at 11:30 | history | asked | Isky Mathews | CC BY-SA 4.0 |