Let M be an
orientablecompactmanifold of dimension n, with boundary , and let 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 with , and of with for all .[2]
Here can in fact be empty, so Poincaré duality appears as a special case of Lefschetz duality.
There is a version for triples. Let 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 , there is an isomorphism[3]
Notes
^Biographical Memoirs By National Research Council Staff (1992), p. 297.
^Vick, James W. (1994). Homology Theory: An Introduction to Algebraic Topology. p. 171.