Does every open orientable even-dimensional smooth manifold admit an almost complex structure?
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2$\begingroup$ Gromov proved that on any open manifold of real dimension less than or equal to 6, almost complex structures exist if and only if complex structures exist. $\endgroup$– Ben McKayCommented Nov 2, 2020 at 8:27
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1$\begingroup$ @BenMcKay: Do you have a reference for this result? $\endgroup$– Michael AlbaneseCommented Nov 2, 2020 at 11:53
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1$\begingroup$ @MichaelAlbanese: Gromov, M. (1973) Convex integration of differential relations, Izv. Akad. Nauk S.S.S.R. 37, # 2, pp. 329-343. $\endgroup$– Ben McKayCommented Nov 2, 2020 at 12:49
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1$\begingroup$ @MichaelAlbanese: also Gromov, Partial Differential Relations, p. 103. $\endgroup$– Ben McKayCommented Nov 2, 2020 at 12:51
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3$\begingroup$ Another interesting related result of Gromov (in his PhD thesis) about open manifolds is that they admit a almost complex structure $\iff$ they admit a symplectic form (very different to the compact case where there is an existance conjecture but it remains open in dimensions $\geq 6$). $\endgroup$– Nick LCommented Nov 2, 2020 at 14:53
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1 Answer
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If $M$ admits an almost complex structure, then the odd Stiefel-Whitney classes vanish and the even Stiefel-Whitney classes admit integral lifts, namely $c_i(M) \equiv w_{2i}(M) \bmod 2$. These two conditions give restrictions on the smooth manifolds which can admit almost complex structures.
The first restriction, namely that $w_1(M) = 0$, is equivalent to orientability. If $M$ is orientable, then the second restriction, namely that $w_2(M)$ admits an integral lift, is equivalent to the manifold being spin$^c$.
An example of an orientable non-spin$^c$ manifold is the Wu manifold $SU(3)/SO(3)$ which has dimension five. Therefore $M = (SU(3)/SO(3))\times\mathbb{R}^{2k+1}$ is an open orientable even-dimensional manifold which does not admit an almost complex structure.
Note that $\dim M = 2k + 6$, so this gives examples in all positive even dimensions other than two and four. It turns out that in dimensions two and four, there are no examples.
- In dimension two, a manifold is almost complex if and only if it is orientable.
- In dimension four, an open manifold admits an almost complex structure if and only if it is spin$^c$, and every orientable four-manifold is spin$^c$, see this note by Teichner and Vogt.
answered Nov 2, 2020 at 11:49
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$\begingroup$ Great answer! I had thought that was the case. I was just reading your paper arxiv-vanity.com/papers/1905.01760 in there it says the connected sum of two closed 4d AC manifolds is never AC (almost complex). This seems contrary to your above statement. I'm sure I'm missing something? NOTE: technically the paper says in dimension 4m m in $\mathbb{Z}$ $\endgroup$ Commented Jan 28, 2023 at 19:39
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$\begingroup$ @R.Rankin: Note that the four-dimensional statement in my answer above only applies to open manifolds, not closed manifolds. $\endgroup$ Commented Jan 28, 2023 at 19:51
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$\begingroup$ Thank you, your work comes up a lot in my "adventures" $\endgroup$ Commented Jan 28, 2023 at 19:58