Does every open orientable even-dimensional smooth manifold admit an almost complex structure?



Does every open orientable even-dimensional smooth manifold admit an almost complex structure?
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](https://math.berkeley.edu/%7Eteichner/Papers/spin.pdf) by Teichner and Vogt.


