Lefschetz duality

Lefschetz–Poincaré duality

An assertion about the duality between homology and cohomology, established by S. Lefschetz. More precisely, if $(X,A)$ is a pair of spaces such that $X\setminus A$ is an $n$-dimensional topological manifold, then for any Abelian group $G$ and any $i$ there is an isomorphism

$$H_i(X,A;G)\approx H_c^{n-i}(X\setminus A;G).$$

On the right-hand side one has cohomology with compact support. If the manifold $X\setminus A$ is non-orientable, one must, as usual, take cohomology with local coefficients.