Is there a simply-connected smooth closed 4-manifold with a characteristic class $x \in H\_2(X; \mathbb{Z})$ such that $x$ can not be represented by a disjoint union of tori in $X$?


I would not know how to prove this without the characteristic hypothesis either so any thoughts on that would also be appreciated.



Characteristic class that cannot be represented by disjoint tori
In $H\_2(CP^2)$, every class $nH$ where $H$ is a generator and n is odd is characteristic. However, if $n >3$, then such a class is not represented by a torus. It is not represented by a disjoint union of tori, either. For non-zero classes in $H\_2(CP^2)$ have non-zero intersection numbers. So if you had a disjoint union of tori, then at most one would be non-trivial in homology (and that one would represent $nH$, contradicting the previous step.)


