Intersection form
= Intersection form
{title2=$Q_M$}
For a <closed manifold> $M$ with an <orientation> and <dimension> $4k$, its integral intersection form is the <symmetric bilinear form> on
$$
L=H_{2k}(M;\mathbb Z)/\operatorname{Tor}H_{2k}(M;\mathbb Z)
$$
obtained by pairing complementary middle-dimensional <homology classes>. Under <Poincare duality>, it is the <cup product> pairing
$$
Q_M(a,b)=\langle a\smile b,[M]\rangle
$$
on degree-$2k$ integral <cohomology> modulo its <torsion subgroup>. The even degree makes the pairing <symmetric>; integral duality makes it a <unimodular intersection pairing>. When representatives are transverse oriented submanifolds, the same value is the signed count of their intersection points.
The <intersection form of a 4-manifold> is the case $k=1$. On the <Complex projective plane> its matrix is $(1)$, whereas on $S^2\times S^2$ the factor spheres give $\begin{pmatrix}0&1\\1&0\end{pmatrix}$.
For a compact <manifold with boundary>, the perfect <intersection pairing> pairs absolute and relative <homology>, $H_{2k}(M)$ with $H_{2k}(M,\partial M)$; the induced absolute self-pairing can be degenerate. For instance the core sphere in $S^2\times D^2$ generates $H_2$ but has self-intersection zero.