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?
MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up.
Sign up to join this communityDoes 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 non-singular quadric, then $Q_n$ has the same integral homology and cohomology groups as $\mathbb{P}^{2n+1}$, but the cohomology rings are different.