Intersection Form and Applications of Poincare Duality

Poincare duality gives rise to the intersection form, a powerful invariant that encodes how submanifolds of complementary dimension intersect within a closed manifold.

The Intersection Form

Definition

Let $M$ be a closed oriented $n$ -manifold with $n = 2m$ . The intersection form is the bilinear pairing $Q_M : H_m(M; \mathbb{Z}) \times H_m(M; \mathbb{Z}) \to \mathbb{Z}$ defined by $Q_M(\alpha, \beta) = \langle D_M^{-1}(\alpha) \smile D_M^{-1}(\beta), [M] \rangle$ , where $D_M$ is the Poincare duality isomorphism. Geometrically, $Q_M(\alpha, \beta)$ counts (with sign) the algebraic intersection number of cycles representing $\alpha$ and $\beta$ in general position.

The intersection form is symmetric when $m$ is even and skew-symmetric when $m$ is odd, as a consequence of the graded-commutativity of the cup product.

Definition

A unimodular (or non-degenerate) form over $\mathbb{Z}$ is a bilinear form whose associated matrix has determinant $\pm 1$ . For a closed oriented manifold $M$ of dimension $2m$ , the intersection form $Q_M$ on $H_m(M; \mathbb{Z}) / \text{torsion}$ is always unimodular by Poincare duality.

Classification in Dimension 4

Theorem8.5Freedman's Classification

Two simply-connected, closed, oriented topological $4$ -manifolds are homeomorphic if and only if their intersection forms are isomorphic over $\mathbb{Z}$ . Every unimodular symmetric bilinear form over $\mathbb{Z}$ is realized by some such manifold.

ExampleIntersection form of $\mathbb{CP}^2$

The complex projective plane $\mathbb{CP}^2$ has $H_2(\mathbb{CP}^2) \cong \mathbb{Z}$ with intersection form $Q = (1)$ , the $1 \times 1$ matrix with entry $1$ . The manifold $\overline{\mathbb{CP}}^2$ (with reversed orientation) has $Q = (-1)$ . The connected sum $\mathbb{CP}^2 \# \mathbb{CP}^2$ has form $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ .

Further Applications

ExampleNon-existence results

Poincare duality constrains which finitely generated abelian groups can be the homology of a closed $4$ -manifold. For instance, no closed oriented $4$ -manifold $M$ can have $H_1(M) = 0$ , $H_2(M) = \mathbb{Z}/3\mathbb{Z}$ , since by duality and the universal coefficient theorem, $H^2 \cong H_2$ would have torsion, but $H_2 \cong H^2$ would need to pair non-degenerately, which $\mathbb{Z}/3\mathbb{Z}$ cannot do over $\mathbb{Z}$ .

RemarkSmooth vs. topological

Donaldson's theorem (1983) shows that the intersection form of a smooth, simply-connected, closed $4$ -manifold is either diagonal $\pm I_n$ (definite case) or one of the standard indefinite forms. This powerful constraint, proved using gauge theory, distinguishes smooth from topological $4$ -manifolds and implies the existence of exotic smooth structures on $\mathbb{R}^4$ .

The intersection form, derived from Poincare duality, sits at the interface of algebraic topology, geometric topology, and mathematical physics, demonstrating how algebraic invariants encode deep geometric information about manifolds.