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 Lefschetz (1926), 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 (co)homology groups of M and the relative (co)homology of the pair (M, N); and this gives rise to isomorphisms of H^k(M, N) with H_{n - k}(M), and of H_k(M, N) with H^{n - k}(M).

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 A and B denote two subspaces of the boundary N, themselves compact orientable manifolds with common boundary Z, which is the intersection of A and B. Then there is an isomorphism

$D_M: H^p(M,A,Z)\to H_{n-p}(M,B,Z).$