The Poincare Duality Theorem

Poincare duality is one of the deepest results in algebraic topology, providing a canonical isomorphism between the cohomology and homology of a closed oriented manifold in complementary dimensions.

The Main Theorem

Theorem8.6Poincare Duality

Let $M$ be a closed, oriented, connected $n$ -manifold. Then for all $0 \leq k \leq n$ , the cap product with the fundamental class $D_M : H^k(M; \mathbb{Z}) \xrightarrow{\;\cong\;} H_{n-k}(M; \mathbb{Z}), \quad \alpha \mapsto \alpha \frown [M]$ is an isomorphism. More generally, for any coefficient ring $R$ , the map $D_M : H^k(M; R) \to H_{n-k}(M; R)$ given by capping with the $R$ -fundamental class is an isomorphism.

The proof requires careful analysis of the local structure of manifolds and proceeds by an inductive argument on the number of cells in a CW decomposition, using the Mayer-Vietoris sequence at each stage.

Poincare Duality for Manifolds with Boundary

Theorem8.7Lefschetz Duality

Let $M$ be a compact oriented $n$ -manifold with boundary $\partial M$ . Then for all $k$ , there are isomorphisms $H^k(M, \partial M; \mathbb{Z}) \cong H_{n-k}(M; \mathbb{Z})$ $H^k(M; \mathbb{Z}) \cong H_{n-k}(M, \partial M; \mathbb{Z})$ given by the cap product with the relative fundamental class $[M, \partial M] \in H_n(M, \partial M)$ .

ExampleLefschetz duality for the disk

For the closed disk $D^n$ with $\partial D^n = S^{n-1}$ , Lefschetz duality gives $H^k(D^n, S^{n-1}) \cong H_{n-k}(D^n) \cong \begin{cases} \mathbb{Z} & k = n \\ 0 & k \neq n \end{cases}$ . This is consistent with the relative cohomology computed via the long exact sequence of the pair.

The Euler Characteristic

Theorem8.8Poincare Duality and the Euler Characteristic

For a closed oriented manifold $M$ of odd dimension $n = 2m+1$ , the Euler characteristic satisfies $\chi(M) = 0$ .

Proof: By Poincare duality, $b_k = b_{n-k}$ . Then $\chi(M) = \sum_{k=0}^n (-1)^k b_k = \sum_{k=0}^n (-1)^{n-k} b_{n-k} = (-1)^n \chi(M) = -\chi(M)$ , so $\chi(M) = 0$ .

RemarkDuality in algebraic geometry

Poincare duality has an algebraic analogue: Serre duality for coherent sheaves on smooth projective varieties. If $X$ is a smooth projective variety of dimension $n$ over a field, then $H^k(X, \mathcal{F}) \cong H^{n-k}(X, \mathcal{F}^\vee \otimes \omega_X)^\vee$ , where $\omega_X$ is the canonical sheaf. This parallel reflects deep connections between topology and algebraic geometry.

Poincare duality is the cornerstone of manifold topology, and its generalizations (Lefschetz duality, Alexander duality, Serre duality) pervade all areas of modern geometry and topology.