Timeline for Why does the longitude correspond to Frobenius in Arithmetic Topology, and other strange phenomena
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 6, 2019 at 22:21 | comment | added | Filippo Alberto Edoardo | Let us continue this discussion in chat. | |
| Jul 6, 2019 at 21:21 | comment | added | Will Sawin | But the idea that $\alpha$ should be the monodromy matches the quote that it should be the generator of the inertia. So what is the confusion? | |
| Jul 6, 2019 at 19:20 | comment | added | Filippo Alberto Edoardo | As for the $\mathbb{Z}_p$-extension, I guess you're right, I have overlooked that when doing the topological parts he sticks to knots rather than links. But do you see any reasonable obstruction to do this for links? Probably, as in the arithmetic case, there would be a failure of codescent? And do you have any feeling about the tameness/wildness problem? | |
| Jul 6, 2019 at 19:18 | comment | added | Filippo Alberto Edoardo | Concerning $\alpha$ vs. $\beta$, remember the relevant Galois groups here are fund'l groups of complements of the neighb'd: for the unknot, for instance, this $\pi_1$ is $\mathbb{Z}$, generated by something which becomes $\alpha$, if I am not mistaken (I am looking at Morishita's ex. 2.6, p. 12). But of course I'd be very happy to be proven wrong and to find again my Frobenius generating the decomposition group! | |
| Jul 6, 2019 at 19:14 | comment | added | Filippo Alberto Edoardo | @WillSawin Well, in considering which loop should be the "right" monodromy, I was thinking at $\alpha$ because I have in mind the idea of a family of curves over the punctured disk (in $\mathbb{C}$, say), or over $\mathbb{Z}_p$; but I agree that this is just a feeling. | |
| Jul 6, 2019 at 16:20 | history | edited | YCor |
edited tags
|
|
| Jul 6, 2019 at 16:01 | comment | added | Will Sawin | Presumably the assumption that the $\mathbb Z_p$-extension be ramified at one prime only is analogous to the knot case, not the link case. Perhaps he is thinking specifically of knots in $S^3$ there? | |
| Jul 6, 2019 at 15:59 | comment | added | Will Sawin | Why can't "a loop going around $K$" be $\beta$? | |
| Jul 6, 2019 at 15:57 | comment | added | Will Sawin | Why does this not agree with the idea that inertia acts as local monodromy, which is running around holds? $D^2$ is the hole, $\partial D^2$ is running around it. | |
| Jul 6, 2019 at 15:38 | history | asked | Filippo Alberto Edoardo | CC BY-SA 4.0 |