Timeline for An almost complex structure on $S^2\times ...\times S^2 / \mathbb{Z_2}$
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Jan 17, 2019 at 12:22 | history | edited | YCor |
edited tags
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| Jan 12, 2019 at 13:00 | vote | accept | aglearner | ||
| Jan 9, 2019 at 6:51 | history | edited | Michael Albanese |
Added almost-complex tag. Tag doesn't exist yet, but I believe it should.
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| Jan 9, 2019 at 5:59 | answer | added | Michael Albanese | timeline score: 20 | |
| Dec 14, 2018 at 16:18 | comment | added | aglearner | Mark and Michael, thanks a lot your comments! | |
| Dec 12, 2018 at 15:16 | comment | added | Mark Grant | Neat. Checking these should be do-able, for someone with sufficient motivation and time on their hands. Note that these manifolds are projective product spaces, and Don Davis has calculated their cohomology and Steenrod operations here: arxiv.org/abs/0908.0525 | |
| Dec 12, 2018 at 15:10 | comment | added | Michael Albanese | @MarkGrant: Theorem 1 of this paper by Heaps completely answers the question of when an eight-manifold has an almost complex structure, but it is significantly more complicated. | |
| Dec 12, 2018 at 15:08 | comment | added | Mark Grant | @MichaelAlbanese: Ah right. I was thinking of $6$-manifolds. What do we know about the obstructions for $8$-manifolds? (This question is relevant, but the answers given only cover dimensions $4$ and $6$: mathoverflow.net/questions/63439/…) | |
| Dec 12, 2018 at 15:03 | comment | added | Michael Albanese | @MarkGrant: That is the first obstruction, but not the only one for a four-manifold. You also need the integral lift $c$ of $w_2$ to satisfy $c^2 = 2\chi + 3\sigma$. Every closed orientable four-manifold has an integral lift of $w_2$ (i.e. they are spin${}^c$), but they do not always admit almost complex structures. | |
| Dec 12, 2018 at 14:59 | comment | added | Michael Albanese | For $n = 1$ we have $X_1 = \operatorname{Gr}^+(2, 4)$ and $X_1/\mathbb{Z}_2 = \operatorname{Gr}(2, 4)$ which does not admit an almost complex structure. | |
| Dec 12, 2018 at 14:28 | comment | added | Mark Grant | For the same reason, this first obstruction to an AC structure vanishes in all the other cases as well. | |
| Dec 12, 2018 at 14:27 | comment | added | Mark Grant | Hmm. I would have said that $X_1/\mathbb{Z}_2$ is almost complex. Since this is an orientable $4$-manifold, the only obstruction is that $w_2$ is the reduction of an integral class. There is a fibration $X_1\to X_1/\mathbb{Z}_2\to B\mathbb{Z}_2$, and $w_2(X_1/\mathbb{Z}_2)$ pulls back to $w_2(X_1)=0$, so comes up from the nonzero class in $H^2(B\mathbb{Z}_2;\mathbb{Z}_2)$, which is an integral reduction. | |
| Dec 12, 2018 at 12:55 | history | asked | aglearner | CC BY-SA 4.0 |