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 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
n
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
{\displaystyle H_{n-k}(M;\mathbb {Z} )}
{\displaystyle H^{n-k}(M;\mathbb {Z} )}
can in fact be empty, so Poincaré duality appears as a special case of Lefschetz duality.
There is a version for triples.
decompose into subspaces A and B, themselves compact orientable manifolds with common boundary Z, which is the intersection of A and B.
, there is an isomorphism[3]