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 $$\partial(M)$$, and let $$z\in H_n(M,\partial(M); \Z)$$ 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 $$(M,\partial(M))$$. Furthermore, this gives rise to isomorphisms of $$H^k(M,\partial(M); \Z)$$ with $$H_{n-k}(M; \Z)$$, and of $$H_k(M,\partial(M); \Z)$$ with $$H^{n-k}(M; \Z)$$ for all $$k$$.

Here $$\partial(M)$$ can in fact be empty, so Poincaré duality appears as a special case of Lefschetz duality.

There is a version for triples. Let $$\partial(M)$$ 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 $$k$$, there is an isomorphism


 * $$D_M\colon H^k(M,A; \Z)\to H_{n-k}(M,B; \Z).$$