= Intersection form of a 4-manifold
{title2=$Q_M(a,b)=\langle a\smile b,[M]\rangle$}
{wiki}
For a <closed manifold> $M$ with an <orientation> and <dimension> four, its intersection form is the <symmetric bilinear form>
$$
Q_M(a,b)=\langle a\smile b,[M]\rangle
$$
on $H^2(M;\mathbb Z)$ modulo its <torsion subgroup>. By <Poincare duality> it is the signed-intersection pairing on the free part of $H_2(M;\mathbb Z)$, and it is a <unimodular intersection pairing>. Its real extension has positive and negative indices whose difference is its <signature>.
This is the four-dimensional case of the general <intersection form>. On a compact oriented four-dimensional <manifold with boundary>, the perfect <intersection pairing> uses one absolute and one relative <homology class>. An absolute self-pairing alone can be degenerate, so it should not be identified with the closed-manifold unimodular form without a boundary qualification.
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