 Intersection form

For a closed manifold $M$ with an orientation and dimension $4k$, its integral intersection form is the symmetric bilinear form onobtained by pairing complementary middle-dimensional homology classes. Under Poincare duality, it is the cup product pairingon 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 $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$.

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.