Page "Poincaré duality" Paragraph 40

This formulation of Poincaré Duality has become quite popular as it provides a means to define Poincaré Duality for any generalized homology theories provided one has a Thom Isomorphism for that homology theory.

A Thom isomorphism theorem for a homology theory is now accepted as the generalized notion of orientability for a homology theory.

For example, a-structure on a manifold turns out to be precisely what is needed to be orientable in the sense of complex topological k-theory.