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?
 A: 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.

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*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.

