Poincaré duality

Remarks

The proof first shows a version with compactly supported cohomology for every $R$-oriented manifold, not only closed ones. It holds for open subsets of $\R^n$ that are convex, and it passes to unions by a Mayer–Vietoris argument and to increasing unions by a limit argument. For a closed manifold, compactly supported cohomology is ordinary cohomology.

Every closed manifold is $\Z/2$-oriented, so the theorem always holds with $\Z/2$ coefficients. With a field $F$ of coefficients, it implies that the pairing $H^k(M; F) \times H^{n-k}(M; F) \to F$, $(a, b) \mapsto \langle a \smile b, [M] \rangle$, is nondegenerate. In particular, the Betti numbers satisfy $b_k(M) = b_{n-k}(M)$ for closed orientable $M$, and the Euler characteristic of a closed manifold of odd dimension is $0$.

The hypotheses are needed. Without orientability: for the real projective plane, $H^0(\R P^2; \Z) = \Z$ but $H_2(\R P^2; \Z) = 0$. Without compactness: $H^0(\R^n; \Z) = \Z$ but $H_n(\R^n; \Z) = 0$ for $n \ge 1$.

Going deeper

For a student

The theorem identifies cohomology and homology of closed oriented manifolds with a shift of degree $n$. It turns the cup product into intersection of cycles, which gives the intersection form of a $4$-manifold and its unimodularity. It yields the symmetry of Betti numbers and the vanishing of the Euler characteristic in odd dimensions. Lefschetz duality extends it to compact manifolds with boundary, and Alexander duality is a consequence for complements of subsets of spheres.