Let $M$ be a simply connected, compact and $\mathbb{Z}$-oriented 4-dimensional manifold. Let $\mu\in H_4(M;\mathbb{Z})$ be a fundamental class of $M$ (here is $H_4(M;\mathbb{Z})$ the singular homology of $M$ with coefficients in $\mathbb{Z}$) and let $$\begin{align} \langle \, , \, \rangle :H^k(M;\mathbb{Z})\otimes H^{4-k}(M;\mathbb{Z}) & \to \mathbb{Z} \\ a\otimes b & \mapsto (a\cup b)(\mu) \end{align}$$ the sectional shape of $M$.
Why does $\langle \, , \, \rangle$ induce an isomorphism $H^2(M;\mathbb{Z})\to Hom(H^2(M;\mathbb{Z}),\mathbb{Z})$?
I don't have a solution but I know a few facts about $M$ which could be helpful (but I don't know)... I know that we have an isomorphism $$\cap \mu :H^p(M;\mathbb{Z})\mapsto H_{4-p}(M;\mathbb{Z})$$ and because $M$ is simply connected, we get $H_3(M;\mathbb{Z})\cong H^1(M;\mathbb{Z})=0$. Therefore the pairing on the (co)-chain complex $C^4(M)\otimes C_4(M)\mapsto \mathbb{Z}$, defined by $\alpha\otimes a\mapsto \alpha (a)$ induces an isomorphism $$H^4(M)\mapsto Hom_\mathbb{Z}(H_4(M),\mathbb{Z}), \psi\mapsto (x\mapsto \psi (x)).$$ My first try was to write $$H^2(M;\mathbb{Z})\to Hom(H^2(M;\mathbb{Z}),\mathbb{Z}),$$ $$\eta\mapsto (\phi \mapsto \langle \eta , \phi\rangle )$$ as a composition of isomorphisms, but I'm stuck. Do you know how to prove the question? Regards
The universal coefficients theorem implies, since $M$ is simply connected (and hence $H_1(M) = 0$), that $$\begin{align} H^2(M) & \to \hom(H_2(M), \mathbb{Z}) \\ \eta & \mapsto (\sigma \mapsto \eta(\sigma)) \end{align}$$ is an isomorphism. Moreover, Poincaré duality implies that $$- \frown \mu : H^2(M) \to H_2(M)$$ is an isomorphism too. So by composing both isomorphisms, we get that $$\begin{align} H^2(M) & \to \hom(H^2(M), \mathbb{Z}) \\ \eta & \mapsto (\phi \mapsto \eta(\phi \frown \mu)) \end{align}$$ is an isomorphism. But the properties of the cap and cup products imply that $\eta(\phi \frown \mu) = \eta \frown (\phi \frown \mu) = (\eta \smile \phi) \frown \mu =: \langle \eta, \phi \rangle$ and so you get your application.