In mathematics, the twisted Poincaré duality is a theorem removing the restriction on Poincaré duality to oriented manifolds.
Another version of the theorem with real coefficients features de Rham cohomology with values in the orientation bundle.
This is the flat real line bundle denoted
As a flat line bundle, it has a de Rham cohomology, denoted by 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, i.e., evaluating against the fundamental class.
Poincaré duality for differential forms is then the conjunction, for M connected, of the following two statements: is non-degenerate.