Inmathematics, Lefschetz duality is a version of Poincaré dualityingeometric topology, applying to a manifold with boundary. Such a formulation was introduced by Solomon Lefschetz (1926), at the same time introducing relative homology, for application to the Lefschetz fixed-point theorem.[1] There are now numerous formulations of Lefschetz duality or Poincaré–Lefschetz duality, or Alexander–Lefschetz 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 groupsofM and the relative (co)homology of the pair
. Furthermore, this gives rise to isomorphisms of
with
, and of
with
for all
.[2]
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[3]