Intersection form

Remarks

$Q_M$ is a symmetric bilinear form, since $a \smile b = b \smile a$ for classes of even degree. It vanishes if $a$ or $b$ is a torsion class, so it induces a form on the free abelian group $H^2(M; \Z)/\text{torsion}$. By Poincaré duality this induced form is unimodular: its matrix in any basis is a unimodular matrix, with determinant $\pm 1$. Its rank is $b_2(M)$, and the signature of its real extension is the signature of $M$.

Example: for the complex projective plane $\C P^2$ with its complex orientation, $H^2 \cong \Z$ is generated by $x$ with $\langle x^2, [\C P^2] \rangle = 1$, so $Q = (1)$. For $S^2 \times S^2$ the form has matrix $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ in the basis given by the two factors.

Non-example: on a closed oriented surface $\Sigma$, the analogous form $H^1(\Sigma) \times H^1(\Sigma) \to \Z$ is skew-symmetric, not symmetric. The reason is that $a \smile b = -\,b \smile a$ in degree $1$. A symmetric form needs dimension $4$, or more generally $4k$ with middle degree $2k$.