Timeline for Rational homotopy groups of a projective hypersurface
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 25, 2018 at 15:14 | answer | added | Aleksandar Milivojević | timeline score: 12 | |
| May 17, 2018 at 20:35 | comment | added | user51223 | @NickL Thanks for correcting me. I need to correct my comment that the isomorphism hold in the stable range, so that if $X$ has its bottom cell in dimension $r+1$ then the rational Hurwicz map is an isomorphism up to dimension $2r$ and an epimorphism in dimension $2r+1$. I have mis-remembered something else which is so embarrassing! | |
| May 16, 2018 at 13:54 | comment | added | user19475 | I found something for étale homotopy groups I can't read: repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/61752/1/… | |
| May 16, 2018 at 13:13 | comment | added | user19475 | For the étale fundamental group: hypersurfaces are almost always simply connected. | |
| May 16, 2018 at 13:11 | comment | added | Nick L | I don't know, sorry. | |
| May 16, 2018 at 13:02 | comment | added | user19475 | Does this also work for étale homotopy groups over arbitrary fields? | |
| May 16, 2018 at 9:33 | comment | added | Nick L | any hypersurface becomes a hyperplane section after a suitable Veronese embedding of the ambient projective space. So by Lefshetz hyperplane theorem for Homotopy groups, for $i<n-1, \pi_{i}(X) $ is isomorphic to $\pi_{i}(\mathbb{CP}^{n})$. These Homotopy groups can be calculated by applying the long exact sequence to the $S^{1}$-bundle $S^{2n+1} \rightarrow \mathbb{CP}^{n}$. | |
| May 16, 2018 at 8:01 | review | First posts | |||
| May 16, 2018 at 8:09 | |||||
| May 16, 2018 at 8:00 | history | asked | H. Tzyn | CC BY-SA 4.0 |