Lefschetz duality

In mathematics, Lefschetz duality is a version of Poincaré duality in geometric topology, applying to a manifold with boundary. Such a formulation was introduced by, at the same time introducing relative homology, for application to the Lefschetz fixed-point theorem. There are now numerous formulations of Lefschetz duality or Poincaré–Lefschetz duality, or Alexander–Lefschetz duality.

Formulations

Let M be an orientable compact manifold of dimension n, with boundary, and let be the fundamental class of the manifold M. Then cap product with z induces a pairing of the homology groups of M and the relative homology of the pair. Furthermore, this gives rise to isomorphisms of with, and of with for all.
Here can in fact be empty, so Poincaré duality appears as a special case of Lefschetz duality.
There is a version for triples. Let decompose into subspaces A and B, themselves compact orientable manifolds with common boundary Z, which is the intersection of A and B. Then, for each, there is an isomorphism