Poincaré Duality in Manifolds and Groups

Poincaré Duality on an orientable n-manifold gives a canonical isomorphism between homology and cohomology. This isomorphism links dimension k with dimension n — k. Ordinary homology is Poincaré dual to cohomology based on finite chains, and ordinary cohomology is Poincaré dual to homology based on infinite chains. The geometric treatment given here exhibits these duality isomorphisms at the level of chains in an intuitively satisfying way. Historically, it is how things were first done. A more sophisticated treatment in which Poincaré Duality is presented as “cap product with a fundamental class” can be found in many modern books on algebraic topology.

We end the chapter with a discussion of Poincaré Duality groups and duality groups.