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 \( M \).
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The intersection form of a 4-manifold is the middle-dimensional signed-intersection pairing of an oriented compact four-manifold. On a closed manifold it can equivalently be defined on second cohomology by evaluating the cup product on the fundamental class. The general intersection form also applies to higher dimensions.