Does every even-dimensional sphere admit an almost complex structure?

Question

We know that there is an almost complex structure on $S^6$ which is not integrable. Is it always possible to find almost complex structures on $S^{2n}$? In particular does $S^4$ admit one?

Answer 1

In Sur les classes caractéristiques des structures fibrés sphériques, Wu proved that $S^{4n}$ cannot admit an almost complex structure for $n \geq 1$.

In Groupes de Lie et puissances réduites de Steenrod, Borel and Serre proved that $S^{2n}$ cannot an admit complex structure for $n \geq 4$.

Therefore, the only even-dimensional spheres which can admit an almost complex structure are $S^2$ and $S^6$. In fact, both of these do admit almost complex structures. The former is the familiar Riemann sphere, while the latter obtains an almost complex structure by viewing it as the unit length purely imaginary octonions. Note, one can obtain the almost complex structure on $S^2$ in an analogous way using quaternions; see this question.

The almost complex structure on $S^2$ is integrable; in fact, every almost complex structure on a two-dimensional manifold is integrable, see this question. However, the almost complex structure on $S^6$ described above is known to be non-integrable. It is still unknown whether $S^6$ admits an integrable almost complex structure. In fact, it is unknown whether there exists a $2n$-dimensional manifold, $n \geq 3$, which admits almost complex structures without admitting an integrable almost complex structure. In dimension $4$, there are examples of manifolds which admit almost complex structures, none of which is integrable; one example is $(S^1\times S^3)\#(S^1\times S^3)\#\mathbb{CP}^2$, see Theorem $9.2$ of Barth, Hulek, Peters, & van de Ven, Compact Complex Surfaces (second edition).

See my answer here for an update on the search for an integrable almost complex structure on $S^6$.