In mathematics, the twisted Poincaré duality is a theorem removing the restriction on Poincaré duality to oriented manifolds. The existence of a global orientation is replaced by carrying along local information, by means of a local coefficient system.
Twisted Poincaré duality for de Rham cohomology
Another version of the theorem with real coefficients features the de Rham cohomology with values in the orientation bundle. This is the flat real line bundle denoted
For M a compact manifold, the top degree cohomology is equipped with a so-called trace morphism
that is to be interpreted as integration on M, ie. evaluating against the fundamental class.
The Poincaré duality for differential forms is then the conjunction, for M connected, of the following two statements:
is non-degenerate.
The oriented Poincaré duality is contained in this statement, as understood from the fact that the orientation bundle o(M) is trivial if the manifold is oriented, an orientation being a global trivialization, ie. a nowhere vanishing parallel section.