Timeline for Universal covering map from $\mathcal{H}$ to $\mathbb{C}\setminus \mathbb{Z}\oplus i\mathbb{Z}$ (the countably punctured complex plane)
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 13, 2013 at 17:59 | vote | accept | Will Chen | ||
| Sep 7, 2013 at 0:17 | comment | added | Maxime Fortier Bourque | The infinite 4-regular tree embeds nicely in the unit disk as the dual graph to the tesselation by regular ideal quarilaterals described by Prof. Eremenko below. This is because the puntured torus is covered by your space $\mathbb{C} \setminus (\mathbb{Z} + i \mathbb{Z})$. Of course, you can lift any deformation-retraction of the punctured-torus onto the wedge of two circles to a deformation-retraction of the unit disk onto the infinite tree. | |
| Sep 6, 2013 at 13:30 | comment | added | Alexandre Eremenko | @oxeimon: I understand that $H$ is equivalent to the unit disc. But what your question means: "$H$ deformation retractable to a tree"? Of course the half-plane or the disc is retractable to a tree... | |
| Sep 5, 2013 at 23:49 | comment | added | Dan Petersen | The tree in part 2 should look something like this: en.wikipedia.org/wiki/File:H2_tiling_24i-1.png | |
| Sep 5, 2013 at 22:59 | comment | added | Will Chen | Alexandre Eremenko: The Riemann mapping theorem shows that the open unit disc is conformally equivalent to the upper half plane. | |
| Sep 5, 2013 at 22:24 | comment | added | Alexandre Eremenko | I did not understand the second question. The universal cover of the punctured torus is the open unit disc. | |
| Sep 5, 2013 at 22:20 | answer | added | Alexandre Eremenko | timeline score: 12 | |
| Sep 5, 2013 at 20:58 | comment | added | Jesse Silliman | Ah, sorry, I should have read more carefully. | |
| Sep 5, 2013 at 20:24 | history | edited | Will Chen | CC BY-SA 3.0 |
clarifications in the title
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| Sep 5, 2013 at 20:24 | comment | added | Will Chen | I actually mean $\mathbb{C}$ minus all the lattice points. | |
| Sep 5, 2013 at 19:57 | history | asked | Will Chen | CC BY-SA 3.0 |
