Timeline for Higher dimensional version of the Hurwitz formula?
Current License: CC BY-SA 4.0
6 events
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| Apr 6, 2022 at 0:50 | history | edited | Will Chen | CC BY-SA 4.0 |
Corrected the definition of $Z$. It used to say $Z = f^{-1}(Y)$.
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| Jul 10, 2019 at 12:52 | comment | added | user45765 | Ah, I see. Thanks a lot for clarification. The essential reasoning is to eliminate euler characteristic of intersection part by codimension 2. Then mayer-vietoris follows easily. | |
| Jul 10, 2019 at 5:26 | comment | added | Jack Huizenga | @user45765: If you cover $X$ by $X - Z$ and a neighborhood of $Z$, then the intersection will be a neighborhood of Z with Z removed. In nice situations this will have the homotopy type of a bundle of spheres over Z, where the dimension of the spheres is one less than the real codimension of Z in X. So if Z has even codimension the spheres have odd dimension, hence Euler characteristic zero. Then the neighborhood of Z has Euler characteristic zero, and your Mayer-vietoris argument will work. | |
| Jun 1, 2019 at 21:48 | comment | added | user45765 | Sorry for a late question. Why $\chi(X-Z)=\chi(X)-\chi(Z)$ requires $Z$ having even real codimension here? Should this not be application of Myer-vietoris type argument here? | |
| Aug 3, 2011 at 16:20 | history | edited | Jack Huizenga | CC BY-SA 3.0 |
added 1 characters in body
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| Aug 3, 2011 at 3:42 | history | answered | Jack Huizenga | CC BY-SA 3.0 |