TITLE: Characteristic class that cannot be represented by disjoint tori
QUESTION [6 upvotes]: 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.

REPLY [15 votes]: 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.)