Let $M$ be a close, connected, orientable manifold.
If $0 \neq \alpha \in H^q (M^n; \Lambda) $ for $q \neq 0,n$ then, is there a $\beta \in H^{n-q}(M^n;\Lambda)$ with $\alpha \cup \beta \neq 0$?
This question arises from Bredon's Topology and Geometry on page 359 Prop 10.1.
Thanks in advance.
Not necessarily. For instance, let $M=\mathbb{R}P^3$ and $\Lambda=\mathbb{Z}$. Then $H^2(M;\Lambda)=\mathbb{Z}/2$ but $H^1(M;\Lambda)=0$, so no such $\beta$ exists for the nontrivial $\alpha\in H^2(M;\Lambda)$.
More generally, for $\Lambda=\mathbb{Z}$, if $\alpha$ is a torsion element of $H^q(M;\Lambda)$, then no such $\beta$ can exist, since $\alpha\cup\beta$ would be a torsion element of $H^n(M;\Lambda)=\mathbb{Z}$ and the only such element is $0$.