Let $S$ be a normal projective complex surface. $S$ is called a rational homology $\Bbb CP^2$ if $H_i(S,\Bbb Q)=H_i(\Bbb CP^2,\Bbb Q)$ for all $i$, and $S$ is called a rational cohomology $\Bbb CP^2$ if $H^*(S,\Bbb Q)=H^*(\Bbb CP^2,\Bbb Q)$. Clearly a rational cohomology $\Bbb CP^2$ is a rational homology $\Bbb CP^2$ by the universal coefficient theorem. Is the converse also true?
Again by the universal coefficient theorem, for $S$ a rational homology $\Bbb CP^2$, we have $H^i(S,\Bbb Q)=H^i(\Bbb CP^2,\Bbb Q)$ for all $i$, but we have to consider the product structure of the cohomology ring. Maybe the condition that $S$ is projective can be useful, but I got stuck.