Timeline for Complex varieties with non-torsion homotopy groups
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
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| Apr 8, 2017 at 12:54 | vote | accept | user43198 | ||
| Apr 8, 2017 at 1:22 | answer | added | Igor Rivin | timeline score: 2 | |
| Apr 8, 2017 at 0:37 | comment | added | Igor Rivin | @MarkGrant Are curves complements of hyperplane arrangements? Because they are all $K(\pi, 1)$ So are products, and fiber products thereof, though I am not sure if there is a classification of which of the latter are algebraic... | |
| Apr 7, 2017 at 23:50 | answer | added | Kevin Casto | timeline score: 11 | |
| Apr 7, 2017 at 17:09 | comment | added | Mark Grant | Further to Donu Arapura's comment, perhaps David Chataur's answer to mathoverflow.net/questions/207448/… can be used to show that the only examples are $K(\pi,1)$'s as mentioned by Piotr Achinger. | |
| Apr 7, 2017 at 15:47 | comment | added | Donu Arapura | A smooth complex variety has the homotopy type of a finite CW complex. So I doubt you'll find any examples with all homotopy groups torsion free. | |
| Apr 7, 2017 at 12:22 | history | edited | YCor | CC BY-SA 3.0 |
clarified the question
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| Apr 7, 2017 at 11:35 | comment | added | user43198 | @SeanTilson Yes. Consider the scheme as a complex manifold (assuming non-singularity). | |
| Apr 7, 2017 at 10:05 | comment | added | Sean Tilson | What do you mean by homotopy group? The ordinary homotopy groups of the complex points? | |
| Apr 7, 2017 at 9:30 | comment | added | Piotr Achinger | There are many interesting $K(\pi, 1)$ examples, e.g. certain complements of hyperplane arrangements. I don't know examples with nontrivial torsion-free higher homotopy groups. | |
| Apr 7, 2017 at 8:26 | comment | added | user43198 | @PiotrAchinger Could you suggest some reference which deals with examples of non-torsion homotopy groups in the complex topology. | |
| Apr 7, 2017 at 8:07 | comment | added | Piotr Achinger | In characteristic $p$, $\pi_q = 0$ for $q>1$ and $\pi_1$ has no $p$-torsion, but that's of course very different from char. $0$ or the complex topology... | |
| Apr 7, 2017 at 7:26 | comment | added | user43198 | @JohnPardon Sorry, meant to write affine scheme | |
| Apr 7, 2017 at 7:25 | history | edited | user43198 | CC BY-SA 3.0 |
added 1 character in body
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| Apr 7, 2017 at 7:12 | comment | added | Sam Nead | Surely you need more hypotheses to make progress? For example, any open subset of $\mathbb{R}^n$ is an affine manifold (all transition maps are the identity). | |
| Apr 7, 2017 at 6:56 | comment | added | Sam Nead | Perhaps what is meant is a manifold where the transition maps are affine? See en.wikipedia.org/wiki/Affine_manifold | |
| Apr 7, 2017 at 2:18 | review | Close votes | |||
| Apr 7, 2017 at 7:45 | |||||
| Apr 7, 2017 at 2:03 | comment | added | John Pardon | Could you clarify what you mean by "affine space"? | |
| Apr 7, 2017 at 1:27 | history | asked | user43198 | CC BY-SA 3.0 |