Timeline for Third cohomology of symplectic $6$-manifolds
Current License: CC BY-SA 3.0
23 events
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| Oct 10, 2020 at 18:57 | comment | added | user164740 | wouldn't you actually get $H_3(N\times S^2, \mathbb{Z})\cong A$? The torsion shifts by 1 from homology to cohomology. | |
| Jun 24, 2017 at 19:13 | comment | added | HJRW | @NickL -- I don't know about complex 3-folds, but Hongbin Sun recently proved the following very strong statement about the abelian groups arising as $H_1$ of a hyperbolic 3-manifold: For any closed hyperbolic 3-manifold $M$ and any fg abelian group $A$, there is a finite-sheeted cover $M_0\to M$ so that $A$ is a summand of $H_1(M_0)$. | |
| Jun 24, 2017 at 17:19 | history | edited | Nick L | CC BY-SA 3.0 |
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| Jun 24, 2017 at 15:48 | history | edited | Nick L | CC BY-SA 3.0 |
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| Jun 24, 2017 at 15:40 | history | edited | Nick L | CC BY-SA 3.0 |
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| Jun 24, 2017 at 15:35 | history | edited | Nick L | CC BY-SA 3.0 |
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| Jun 24, 2017 at 15:27 | history | edited | Nick L | CC BY-SA 3.0 |
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| Jun 24, 2017 at 13:37 | history | edited | Nick L | CC BY-SA 3.0 |
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| Jun 24, 2017 at 13:26 | comment | added | Nick L | @Ali Taghavi this is proposition 8 of "cubic forms and complex three-folds" by Okonek and Van de Ven (I think it originates back to Wall, but I am not certain of this) | |
| Jun 24, 2017 at 13:05 | comment | added | Nick L | @Ali Taghavi No for any orientable $6$-manifold the intersection form on $H^{3}(M,\mathbb{R})$ is symplectic so $b_{3}$ is even | |
| Jun 24, 2017 at 11:59 | comment | added | Ali Taghavi | @NickL Is there an example of odd rank? | |
| S Jun 24, 2017 at 11:56 | history | suggested | Ali Taghavi |
I add a tag
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| Jun 24, 2017 at 11:55 | comment | added | Ali Taghavi | @NickL What is a reference to the last part of your question"existence of an almost complex structure"? | |
| Jun 24, 2017 at 11:42 | review | Suggested edits | |||
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| Jun 24, 2017 at 11:40 | comment | added | Ali Taghavi | @NickL If $A$ is torsion free of rank $2g$ then $\Sigma_{g} \times \mathbb{C}P^2$ is the desired manifold. | |
| Jun 23, 2017 at 22:54 | history | edited | YCor |
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| Jun 23, 2017 at 21:07 | history | edited | Nick L | CC BY-SA 3.0 |
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| Jun 23, 2017 at 19:45 | history | edited | Nick L |
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| Jun 23, 2017 at 19:32 | comment | added | Nick L | Thanks for your comment! in what sense can we produce more torsion with these examples than say complex projective 3-folds? | |
| Jun 23, 2017 at 18:46 | comment | added | David Treumann | If your goal is to find examples with "lots" of torsion in H^3, instead of a specific group, Reznikov showed the twistor bundle over a hyperbolic 4-manifold is symplectic. | |
| Jun 23, 2017 at 18:07 | history | edited | Nick L | CC BY-SA 3.0 |
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| Jun 23, 2017 at 18:00 | history | edited | Nick L | CC BY-SA 3.0 |
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| Jun 23, 2017 at 17:54 | history | asked | Nick L | CC BY-SA 3.0 |