$\mathbf {The \ Problem \ is}:$ Let $M$ be an $\mathbb{F}$-oriented manifold of dimension $2 n$ for a field $\mathbb{F}$. Consider the non-singular bilinear form $H^{n}(M ; \mathbb{F}) \otimes H^{n}(M ; \mathbb{F}) \rightarrow \mathbb{F}$ given by $$ \langle\alpha, \beta\rangle=\langle\alpha \cup \beta,[M]\rangle . $$ a) Prove that this form is symmetric if $n$ is even, and skew symmetric if $n$ is odd.
b) Classify all symmetric non-degenerate bilinear forms (up to isomorphism) on a $\mathbb{Z} / 2$-vector space $V$ of dimension $\leq 2$.
Find oriented 4-manifolds such that the underlying form is the given one in each case for $\mathbb{F}=\mathbb{Z} / 2$.
$\mathbf {My \ approach}:$ If $\operatorname{dim}(V)=1$, then a symmetric, bi-linear form is given by $\langle a, b\rangle=(a+b)\langle e, 0\rangle$ where $\langle e\rangle=V$, So, a non-trivial bi-linear form is $\langle a, b\rangle=a+b$ and it's non - degenerate.
If $\operatorname{dim}(V)=2$, then if the bi-linear form is given by $\langle(a, b),(c, d)\rangle=ac+bd$ is symmetric and non-degenerate .
I found in a book that upto isometry, this is the only possible symmetric non-degererate, bi-linear form if $dim(V)=2.$
Take $S^{4}$ then $H^{2}\left(S^{4} ; \mathbb{Z}_{2}\right)=0$ hence $\alpha \cup \beta=0 \in H^{4}\left(S^{4} ; \mathbb{Z}_{2}\right)$.
Take $\mathbb{CP}^{2}$, then if $\alpha$ generates $H^2(\mathbb{CP}^{2})$ implies $\alpha^{2}$ generates $H^{4}\left(\mathbb{CP}^{2}\right)$ $\therefore H^{2}\left(\mathbb {CP}^{2};\mathbb Z_{2}\right) \otimes H^{2}\left(\mathbb{C P} ; \mathbb{Z}_{2}\right) \rightarrow \mathbb{Z}_{2}$ is given by $\langle {\alpha}^{2}, [\mathbb {CP}^{2}]\rangle=1$ as ${\alpha}^{2}$ is dual of $[\mathbb {CP^2}].$
I can't find any other compact,orientable $4-$ manifolds . A help is much needed,thanks in advance .