Lefschetz duality explained

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.^[1] 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 .^[2]

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^[3] D _{M\colon} H ^{k(M,A;} \Z)\to H _{n-k} (M,B; \Z).

Notes

1. Biographical Memoirs By National Research Council Staff (1992), p. 297.
2. Book: Vick, James W.. Homology Theory: An Introduction to Algebraic Topology. 1994. 171.
3. Book: Hatcher, Allen. Allen Hatcher. Algebraic topology. Cambridge University Press. Cambridge. 2002. 0-521-79160-X. 254.