For a closed manifold with an orientation and dimension four, its intersection form is the symmetric bilinear formon modulo its torsion subgroup. By Poincare duality it is the signed-intersection pairing on the free part of , 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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In differential topology, the intersection form of a 4-manifold is an important algebraic invariant that captures information about how surfaces intersect within the manifold. Specifically, consider a smooth, closed, oriented 4-manifold \( M \). The intersection form is defined using the homology of \( M \).