Timeline for Motivation of the fundamental theorem of covering spaces
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 11, 2022 at 17:22 | comment | added | Bma | @user481980 Historically, the theory of covering spaces was developed as an application to complex analysis, in the effort to extend the results of Riemann to the full uniformization theorem. So this example is not originally an application of the theory of covering spaces, because it predates them and was the chief inspiration for their definition. Covering spaces were initially an abstraction designed to encapsulate this example for the purposes of a larger problem. See people.dm.unipi.it/benedett/AbikoffUNIF.pdf | |
| May 11, 2022 at 12:35 | comment | added | user481980 | @Bma Your first comment sounds like it is a nice phenomenon that was discovered by looking at a specific example, rather than something that's proved in ordered to have applications elsewhere. Correct? | |
| May 9, 2022 at 23:44 | history | became hot network question | |||
| May 9, 2022 at 19:53 | comment | added | Bma | It’s also worth noting that the covering theory of compact Riemann surfaces corresponds directly to the Galois theory of their function fields. If $\pi: X \to Y$ is a holomorphic cover, the pullback of the function field of $K(Y)$ via $\pi$ makes $K(X)$ a finite degree field extension, and the automorphism group of the extension is the group of deck transformations, with intermediate fields corresponding to intermediate covers (at least when Galois). I do not know who first observed this, but it seems like Riemann would have had the tools to do so. See Förster’s textbook for details. | |
| May 9, 2022 at 19:40 | answer | added | Jonny Evans | timeline score: 8 | |
| May 9, 2022 at 19:14 | comment | added | Bma | I am not a historian, but I think part of the original motivation came from Riemann and analytic continuation. The analytic continuation of say, the logarithm comes equipped with an obvious covering map. Anyone could observe that the homotopy class of a loop in $\mathbb{C}^\times$ corresponds uniquely to a numerical invariant (the index of the curve at 0), to a preimage of the basepoint via the cover, and ultimately to a deck transformation. One could also observe that the analytic continuation of the square root has a smaller deck group. So the fundamental theory is there. | |
| May 9, 2022 at 18:11 | answer | added | Maxime Ramzi | timeline score: 4 | |
| May 9, 2022 at 17:57 | answer | added | Will Sawin | timeline score: 7 | |
| May 9, 2022 at 17:46 | comment | added | user481980 | @AndreasBlass Posets are more down-to-earth than categories. But honestly, "more down-to-earth" is just a phrase I used and I care too less about it to argue about it. This is really not the point of my question. | |
| May 9, 2022 at 17:44 | comment | added | user481980 | @WillSawin "Do you want an explanation of why we should expect that such a result is true, or why we should want to prove such a result and find it useful?" -> why we should want to prove such a result and find it useful, but the first would also be interesting. I am familiar with classical Galois theory. | |
| May 9, 2022 at 17:09 | comment | added | Andreas Blass | I"m not convinced that conjugacy classes of subgroups of $G$ are "more down-to-earth" than left $G$-sets. Subgroups of $G$ are just the stabilizers of points in $G$-sets, and two subgroups are conjugate iff they're the stabilizers of two points in the same orbit. | |
| May 9, 2022 at 16:34 | review | Close votes | |||
| May 15, 2022 at 23:34 | |||||
| May 9, 2022 at 16:10 | comment | added | Will Sawin | "Motivation" means a couple different things and it would help to know more precisely what you mean. Do you want an explanation of why we should expect that such a result is true, or why we should want to prove such a result and find it useful? Another question - are you familiar with the analogous statements in classical Galois theory (inverse isomorphism of posets between subgroups of the Galois group and field extensions, or a similar equivalence of categories)? | |
| May 9, 2022 at 15:44 | comment | added | Wojowu | See here for some applications of etale fundamental groups: mathoverflow.net/q/386011/30186 | |
| May 9, 2022 at 15:40 | history | asked | user481980 | CC BY-SA 4.0 |