Cap Product and Poincare Duality

The cap product is the algebraic operation underlying Poincare duality. It pairs cohomology with homology to produce homology in complementary dimension.

The Cap Product

Definition

The cap product is a bilinear pairing $\frown \;: H^k(X; R) \times H_n(X; R) \to H_{n-k}(X; R)$ defined at the chain level as follows. For a cochain $\varphi \in C^k(X; R)$ and a singular $n$ -simplex $\sigma : \Delta^n \to X$ , $\varphi \frown \sigma = \varphi(\sigma|_{[v_0, \ldots, v_k]}) \cdot \sigma|_{[v_k, \ldots, v_n]}$ where $\sigma|_{[v_0,\ldots,v_k]}$ is the front $k$ -face and $\sigma|_{[v_k,\ldots,v_n]}$ is the back $(n-k)$ -face. This descends to homology/cohomology classes.

The cap product satisfies the fundamental identity relating it to the cup product:

$\langle \alpha \smile \beta, \sigma \rangle = \langle \alpha, \beta \frown \sigma \rangle$

where $\langle -, - \rangle$ denotes the Kronecker pairing between cohomology and homology.

Definition

The Kronecker pairing (or evaluation map) is $\langle -, - \rangle : H^n(X; R) \times H_n(X; R) \to R, \quad \langle [\varphi], [\sigma] \rangle = \varphi(\sigma)$ This is well-defined since cocycles vanish on boundaries and coboundaries vanish on cycles.

The Duality Isomorphism

Theorem8.3Poincare Duality

Let $M$ be a closed, oriented, connected $n$ -manifold with fundamental class $[M] \in H_n(M; \mathbb{Z})$ . Then the map $D_M : H^k(M; \mathbb{Z}) \to H_{n-k}(M; \mathbb{Z}), \quad \alpha \mapsto \alpha \frown [M]$ is an isomorphism for all $0 \leq k \leq n$ .

ExamplePoincare duality for surfaces

For a closed oriented surface $\Sigma_g$ of genus $g$ , Poincare duality gives isomorphisms $H^0 \cong H_2 \cong \mathbb{Z}$ , $H^1 \cong H_1 \cong \mathbb{Z}^{2g}$ , and $H^2 \cong H_0 \cong \mathbb{Z}$ . The intersection form on $H_1(\Sigma_g)$ is the standard symplectic form with matrix $\begin{pmatrix} 0 & I_g \\ -I_g & 0 \end{pmatrix}$ .

Consequences

RemarkSymmetry of Betti numbers

An immediate consequence of Poincare duality is that the Betti numbers of a closed oriented $n$ -manifold satisfy $b_k = b_{n-k}$ . Equivalently, the Poincare polynomial satisfies $P(t) = t^n P(1/t)$ . This constrains which groups can appear as homology of closed manifolds and is one of the first obstructions in manifold recognition.

Poincare duality transforms cohomological operations into geometric ones: cup products become intersection products, and the algebra of cohomology encodes the geometry of intersecting submanifolds.