Almost complex structure question

Question

I know that if $M$ admits an almost complex structure $J$, then $\text{dim}_{\mathbb{R}}(M)=2k$, thus every odd-dimensional manifold doesn't admit an almost complex structure. My question is, are there even-dimensional, orientable manifolds that don't admit an almost complex structure?

Answer 1

Yes, there are many.

Theorem (Borel-Serre) The only spheres $S^n$, $n > 0$, that admit an almost complex structure are $S^2$ and $S^6$.

As you surely know, $S^2$ admits a complex structure, and the sphere endowed with that structure is usually called the Riemann sphere. Whether $S^6$ admits a complex structure is an open question (but see, e.g., this thread on MathOverflow); the usual almost complex structure on $S^6$, inherited from the octonion algebra, is nonintegrable.

Borel, A., Serre, J.-P. "Groupes de Lie et puissances réduites de Steenrod." Amer. J. Math. 75 (1953), 409–448.

A modern, self-contained presentation of a proof can be found in these recent talk notes.