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