On a closed orientable manifold, cap product with the fundamental class exchanges degree $k$ cohomology with degree $n-k$ homology.Suggest a slogan
Let \(M\) be a closed orientable \(n\)-manifold and \(R\) a commutative ring; for example \(R = \ZZ\) or \(R\) a field. Then cap product with the fundamental class gives isomorphisms \[ H^{k}(M; R) \xrightarrow{\ \sim\ } H_{n-k}(M; R) \] for all \(k\) [@Hat02].
For \(n\geq 1\), the orientable manifold \(\RR^n\) is not closed, and \(H^0(\RR^n;\ZZ)\cong\ZZ\) while \(H_n(\RR^n;\ZZ) = 0\). The closed manifold \(\RP^2\) is not orientable, and \(H^0(\RP^2;\ZZ)\cong\ZZ\) while \(H_2(\RP^2;\ZZ) = 0\). For a field \(F\) of characteristic \(2\), every closed manifold is \(F\)-orientable, so the duality holds for all closed manifolds with \(\ZZ/2\) coefficients.