I am studying for a qualifying exam, and I came across this problem:
Let $M$ be a closed orientable connected 4-manifold with $H^1(M) = H^3(M) = 0$ and $H^2(M) \cong H^4(M) \cong \mathbb Z$. What are the possible cup product structures on $H^*(M)$?
My thoughts: Just using properties of graded rings, we see that the product will be determined by $\alpha^2 \in H^4$, where $\alpha$ is a generator of $H^2$. If we fix $\beta$ a generator of $H^4$, then $\alpha^2 = n\beta$, so it seems like there is a distinct structure for each $n \in \mathbb Z$.
My question is: is there anything special about the cohomology ring $M$ that restricts the possibilities further? Somehow my answer seems too easy.