Poincare Duality and Compactly Supported Cohomology

Poincare duality is one of the most profound results in topology, establishing a symmetry in the cohomology of oriented closed manifolds. Its proof in the de Rham setting uses integration to construct a non-degenerate pairing.

Compactly Supported Cohomology

Definition5.5Compactly supported de Rham cohomology

Let $\Omega^k_c(M)$ denote the space of compactly supported smooth $k$ -forms on $M$ . The compactly supported de Rham cohomology is

H^k_c(M) = \frac{\ker(d: \Omega^k_c \to \Omega^{k+1}_c)}{\operatorname{im}(d: \Omega^{k-1}_c \to \Omega^k_c)}.

For compact manifolds, $H^k_c(M) = H^k_{\mathrm{dR}}(M)$ . For non-compact manifolds, they differ.

ExampleCompactly supported cohomology of $\mathbb{R}^n$

$H^k_c(\mathbb{R}^n) \cong \mathbb{R}$ if $k = n$ and $0$ otherwise. The generator of $H^n_c(\mathbb{R}^n)$ is any bump $n$ -form with total integral 1. This contrasts with $H^k_{\mathrm{dR}}(\mathbb{R}^n) = 0$ for $k \geq 1$ .

The Poincare Duality Pairing

Definition5.6Integration pairing

For a closed oriented $n$ -manifold $M$ , the integration pairing is the bilinear map

\langle \cdot, \cdot \rangle: H^k(M) \times H^{n-k}(M) \to \mathbb{R}, \quad \langle [\alpha], [\beta] \rangle = \int_M \alpha \wedge \beta.

This is well-defined: if $\alpha' = \alpha + d\eta$ , then $\int_M \alpha' \wedge \beta = \int_M \alpha \wedge \beta + \int_M d(\eta \wedge \beta) = \int_M \alpha \wedge \beta$ by Stokes' theorem (since $M$ is closed).

Theorem5.4Poincare duality

If $M$ is a closed oriented $n$ -manifold, the integration pairing

H^k_{\mathrm{dR}}(M) \times H^{n-k}_{\mathrm{dR}}(M) \to \mathbb{R}

is non-degenerate. In particular, $H^k(M) \cong (H^{n-k}(M))^*$ , and if the cohomology is finite-dimensional, $\dim H^k(M) = \dim H^{n-k}(M)$ .

Consequences

ExampleSymmetry of Betti numbers

For a closed oriented $n$ -manifold, the Betti numbers satisfy $b_k = b_{n-k}$ . For example:

$S^2$ : $b_0 = b_2 = 1$ , $b_1 = 0$ .
$T^2$ : $b_0 = b_2 = 1$ , $b_1 = 2$ .
$\mathbb{C}P^2$ : $b_0 = b_4 = 1$ , $b_2 = 1$ , $b_1 = b_3 = 0$ .

RemarkPoincare duality for manifolds with boundary

For compact oriented manifolds with boundary, Poincare duality takes the form $H^k(M) \cong (H^{n-k}(M, \partial M))^*$ , where $H^*(M, \partial M)$ denotes relative cohomology (forms vanishing on $\partial M$ ). This is the Lefschetz duality theorem.

Definition5.7Euler characteristic via Poincare duality

The Euler characteristic $\chi(M) = \sum_{k=0}^n (-1)^k b_k$ satisfies $\chi(M) = 0$ for odd-dimensional closed oriented manifolds (since $b_k = b_{n-k}$ and the terms cancel in pairs). For even-dimensional manifolds, $\chi$ is a fundamental topological invariant.

RemarkIntersection form

On a closed oriented $4$ -manifold $M^4$ , Poincare duality restricts to a non-degenerate symmetric bilinear form on $H^2(M; \mathbb{Z})$ , the intersection form. Freedman and Donaldson showed that this form determines the homeomorphism type (simply connected case) but not the diffeomorphism type, leading to the discovery of exotic smooth structures on $\mathbb{R}^4$ .