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 PoincaréâÂÂLefschetz duality, or AlexanderâÂÂLefschetz duality.
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 (or its dual class in cohomology) induces a pairing of the (co)homology groups of M and the relative (co)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