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).

Notes

References

References

1. Biographical Memoirs By National Research Council Staff (1992), p. 297.
2. Vick, James W.. (1994). "Homology Theory: An Introduction to Algebraic Topology".
3. Hatcher, Allen. (2002). "Algebraic topology". [[Cambridge University Press]].