Timeline for Fundamental group of the grid on $\mathbb{R}^\mathbb{N}$
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 4 at 23:10 | comment | added | Corentin B | My bad, I got way too confident | |
| Dec 4 at 19:40 | comment | added | Andy Putman | @HJRW: The sequence $(e_n)$ doesn't just have a convergent subsequence, it actually converges to $0$. | |
| Dec 4 at 19:02 | vote | accept | Dominic van der Zypen | ||
| Dec 4 at 18:45 | answer | added | Jeremy Brazas | timeline score: 8 | |
| Dec 4 at 17:33 | comment | added | HJRW | It may be that, in order to think about this group, one needs to use the techniques referenced in the answers to this MSE question: math.stackexchange.com/questions/667035/… . | |
| Dec 4 at 17:26 | comment | added | HJRW | @CorentinB: It's natural to suspect that it's a graph, but I don't think it is. The product topology is different from the weak topology. For instance, let $e_n$ be the $n$th unit vector. The sequence $(e_n)$ has no convergent subsequence in the weak topology. But $[0,1]^{\mathbb{N}}$ is compact by Tychonoff's theorem, so the sequence does have a convergent subsequence in the product topology. Now, it may still be an increasing union of graphs, and that may be enough to get the conclusion, but it doesn't follow immediately, I think. | |
| Dec 4 at 16:52 | comment | added | Jeremy Brazas | @CorentinB $X$ is path connected because $X$ has the product topology you can create infinite concatenations of paths that move in one coordinate at a time. | |
| Dec 4 at 16:20 | comment | added | Corentin B | I mean, this is a simplicial graph, and $\pi_1$ of graphs are free. | |
| Dec 4 at 16:19 | comment | added | Corentin B | $X$ is not path-connected, and any path-connected component is a increasing union of grids in finite dimension, $\pi_1(X)$ is just going to be a free group of countably infinite rank (as for the grid in $\mathbb R^2$). | |
| Dec 4 at 15:02 | comment | added | Moishe Kohan | Did you think through the case of a grid in $\mathbb R^4$? What group did you get in this case? | |
| Dec 4 at 14:42 | history | asked | Dominic van der Zypen | CC BY-SA 4.0 |