Timeline for Riemann-Hurwitz for real maps
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Dec 19, 2019 at 12:04 | history | edited | Wojowu | CC BY-SA 4.0 |
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| Dec 19, 2019 at 11:38 | history | edited | aglearner | CC BY-SA 4.0 |
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| Dec 19, 2019 at 8:49 | vote | accept | aglearner | ||
| Dec 19, 2019 at 2:42 | answer | added | Will Sawin | timeline score: 4 | |
| Dec 19, 2019 at 0:26 | comment | added | aglearner | I see. In this case, the map is a topological ramified cover, so indeed this formula works. I missed that this condition was imposed. But my question is about generic maps, which will, in particular, change the orientation from time to time. | |
| Dec 19, 2019 at 0:19 | comment | added | Will Sawin | Oh sorry, I was keeping the assumption "only finitely many singularities" from my first comment. | |
| Dec 19, 2019 at 0:07 | history | edited | aglearner | CC BY-SA 4.0 |
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| Dec 19, 2019 at 0:04 | comment | added | aglearner | Will, this formula with the winding number can't work. Consider the map $(x,y)\to (x^2, y)$ then the contribution of each point with $x=0$ is equal to $-1$ according to the formula you propose... | |
| Dec 18, 2019 at 23:21 | comment | added | aglearner | But 1) folds are not isolated. 2) consider the map $z\to z^2$ and perturb it by a real linear map. The resulting map will have three cusps. Do you say that each of them contribute $\pm 1$? (I don't say this is not so, just curious). Anyway, do you think you can prove what you write in the comment? | |
| Dec 18, 2019 at 23:12 | comment | added | Will Sawin | If $x$ is a singularity of $f$, then a loop around $x$ will be sent to some kind of curve in a small neighborhood of $f(x)$, which for a sufficiently small loop and an analytic map won't touch $f(x)$. The local term should be the winding number minus one. | |
| Dec 18, 2019 at 22:54 | comment | added | aglearner | Dear Will, thanks for this comment. Let's even assume for simplicity that the map is analytic or has only simple singularities (in the sense of Arnold). Do you have an idea of what will be the formula? | |
| Dec 18, 2019 at 21:57 | comment | added | Will Sawin | Surely yes if the map has finitely many singularities, so we can compute the Euler class of the of the tangent bundle of $S$ in terms of the pullback of the Euler class of the tangent bundle of $\mathbb P^1$ and local contributions at singularities. | |
| Dec 18, 2019 at 18:40 | history | edited | aglearner |
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| Dec 18, 2019 at 18:13 | history | edited | aglearner | CC BY-SA 4.0 |
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| Dec 18, 2019 at 17:39 | history | asked | aglearner | CC BY-SA 4.0 |