Let $M$ be an $R$-orientable, closed manifold of dimension $2n$ for some $n$. I read in a few texts that the cup product $$H^n(M;R)\times H^n(M;R)\to H^{2n}(M;R)$$ is non degenerate, as a result of the Poincaré duality. However, I fail to see why this is the case. It is equivalent to proving that the cap product $$H^n(M;R)\times H_n(M;R)\to H_0(M;R)$$ is non degenerate. Unfortunately I also fail to see this.
Can anyone help me out?
Let $[M]$ be the fundamental class, the Poincare dualtiy defines an isomorphism:
$H^n(M)\rightarrow H_n(M)$ by $\alpha\rightarrow p(\alpha)=[M]\cap \alpha$ where $\cap $ is the cap product.
If $\alpha$ and $\beta$ are $n$-forms, The cup product $(\alpha\cup\beta)[M] =\alpha( p(\beta))$ (see the reference) suppose that $\alpha\cup\beta=0$ for every $\beta$, you deduce that $\alpha(p(\beta))=0$ for every $\beta$ since $p$ is an isomorphism, $\alpha=0$. Henceforth the cup prodcut is not degenerated.
https://en.wikipedia.org/wiki/Cap_product#Equations