@MichaelAlbanese: Wasn't that yesterday's question? In any case, yes: put the basepoint in different components. I guess Ted just gave the non-stupid answer, though.
@TedShifrin: Do you know if there is a classification of compact almost complex surfaces? By an almost complex surface, I mean a four-dimensional manifold with an almost complex structure.
The integrable case is the Kodaira-Enriques classification.
I trying to become a better mathematician by mastering the art of proof. I've taken lots of plug-n-chug courses. Is their any basic theorem that I could prove?
@MichaelA: Let $M$ be a 4-manifold with fundamental group $G$. If $w_2$ has an integral lift (that is, $\beta(w_2) = 0$), you should be able to connect sum with an appropriate simply connected 4-manifold to obtain an almost-complexable 4-fold with given fundamental group.
So the question is how to build $M$ so that $\beta(w_2) = 0$.
Yes, that's correct. Perhaps they like the finiteness constraint because they can just pass to $(f_1 \cdots f_n)^{-1}(0)$ instead of thinking about intersections of closed sets or whatever.