Timeline for "Cute" applications of the étale fundamental group
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Dec 30, 2021 at 3:41 | comment | added | user149000 | @Libli in my opinion this is definitely something you can teach in a first-year grad course in AG if you're planning to introduce etale $\pi_1$. The result may seem quite out-of-the-blue at first glance, but the proof isn't very advanced. It follows quite readily from the homotopy exact sequence, which is definitely a key feature of etale $\pi_1$ you would probably want to include. All of this and more is also included in Tamas's book (in Gabriel's answer) which, despite Martin's points, I would still heartily recommend. | |
| Dec 30, 2021 at 3:36 | comment | added | user149000 | The action is actually quite easy to describe if you accept the homotopy exact sequence for etale fundamental groups, which splits up $\pi_1(X)$ (where $X$ is the rational projective line minus 3 points) into its geometric component, which is just the profinite completion of $F_2$, and its arithmetic part, which is the Galois group. The existence of a rational point on $X$ gives a section by functoriality, so there's a map back from the Galois group to $\pi_1(X)$. You use this to conjugate $\widehat{F_2}$, so you get an outer action. | |
| Dec 30, 2021 at 3:31 | comment | added | user149000 | @coudy not even close... you aren't staying in $\overline{\mathbb{Q}}$ even if you restrict to piecewise linear loops, the action of this Galois group is totally independent of the Euclidean topology on $\mathbb{C}$ (which you need for continuity), even if all that were true there's no reason homotopic loops should remain homotopic, etc... | |
| Dec 30, 2021 at 3:14 | comment | added | user149000 | @Kimball there doesn't seem to be very good evidence that this is the case; see the papers/surveys of Leila Schneps on this topic. | |
| Dec 13, 2021 at 20:14 | comment | added | coudy | I am confused. Isn't the action of $Gal(\bar{\bf Q}/{\bf Q})$ the obvious one? Since $\sigma(0)=0$ and $\sigma(1)=1$, it acts on $\bar{\bf Q}-\{0,1\}$ and thus also on the space of loops, say piecewise linear ones if we want to stay in $\bar{Q}$, and that space is the free group (up to conjugacy because we miss a basepoint). Take a completion if needed. | |
| Jul 1, 2021 at 8:48 | comment | added | Martin Brandenburg | Is there a nice application of this embedding which only talks to algebraic extensions of $\mathbb{Q}$? | |
| May 2, 2021 at 21:35 | vote | accept | Libli | ||
| Mar 10, 2021 at 18:01 | comment | added | Kimball | which sometimes is conjectured to be isomorphic to Gal(Q¯¯¯¯/Q) - sometimes conjectured, as in sometimes it's conjectured to not be isomorphic also? | |
| Mar 10, 2021 at 9:02 | history | edited | Glorfindel | CC BY-SA 4.0 |
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| Mar 10, 2021 at 7:36 | comment | added | Libli | This is certainly a very nice example and I agree I ave absolutely no idea on how I may possibly prove this result from scratch. I was howeer thinking of applications I may explain to grad students for which I teach a first (slightly advanced) course in Algebraic Geometry. I will edit my question. | |
| Mar 10, 2021 at 1:40 | review | First posts | |||
| Mar 10, 2021 at 2:58 | |||||
| Mar 10, 2021 at 1:37 | history | answered | user175608 | CC BY-SA 4.0 |