Poincaré duality

The Poincaré duality, named after Henri Poincaré, is in algebraic topology is a fundamental relationship between the homology and cohomology of orientable manifolds.

Statement

Be an n-dimensional closed orientable manifold and a natural number, then the k-th singular cohomology group is isomorphic to the ( n - k )-th singular homology group. The isomorphism is realized by the cap product with the fundamental class.

In particular, this applies to the Betti numbers.

History

The identity was first asserted in 1893 by Poincaré. In 1895, he gave a proof in Analysis Situs, where he Betti numbers first through chains of submanifolds (rather than, as in his later works on chains of simplices ) defined and used to prove average numbers of submanifolds. In the addenda to Analysis Situs he defined homology as simplicial homology triangulated manifolds (but not discussed its independence from the triangulation ) and then today was the usual proof of the duality theorem on dual triangulations.

Smooth manifolds

If the manifold additionally smooth, then there is in addition to the singular cohomology and the De Rham cohomology. By the theorem of de Rham cohomology of the corresponding singular and de Rham cohomology groups are isomorphic. With the space of k- differential forms is called. The Hodge star operator

Induces an isomorphism for each between the de Rham cohomology groups. The following diagram

Commutes so.