what are fundamental groups of (almost) complex manifolds?

Question

Are there any restrictions on the fundamental group of an even-dimensional manifold admitting an almost complex structure? integrable almost complex structure? or can I construct any of these with a given fundamental group (presumably finitely generated)?

Answer 1

I assume all manifolds are closed (and hence that we're talking about finitely presented fundamental groups).

There are no restrictions for an almost complex structure. The notion of almost complex structure is very generous; for 4-manifolds it's an entirely algebraic condition that's not hard to satisfy. See here. To do so, just take your favorite smooth compact manifold with chosen fundamental group and connect sum with a few copies of $\Bbb{CP}^2$; you eventually get something with an almost complex structure. (Prove this.) If you're not feeling it, use Gompf's (much harder!) theorem that you can realize every finitely presentable group as the fundamental group of a closed symplectic 4-fold.

For compact complex 3-folds you may have whatever finitely presentable fundamental group you desire. This is known as Taubes's theorem. See Francesco Polizzi's answer here.

You cannot realize every finitely presentable group as the fundamental group of a compact complex surface as a corollary of the Kodaira classification. See here; one example provided is that the free group on $r$ generators, $r>1$ odd, cannot be the fundamental group of a compact complex surface.