Does anyone have an example of two spaces which have the same homology groups, the same cohomology groups, but have different cohomology rings?
is it possible?
Does anyone have an example of two spaces which have the same homology groups, the same cohomology groups, but have different cohomology rings? is it possible? 


A very standard example would be $S^2\vee S^4$ and $\mathbf{C}P^2$. 


There are also standard examples in which both spaces are compact manifolds. For instance, if $n \geq 1$ is an integer and $Q_n \subset \mathbb{P}^{2n+2}$ is a nonsingular quadric, then $Q_n$ has the same integral homology and cohomology groups as $\mathbb{P}^{2n+1}$, but the cohomology rings are different. 

