# 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.