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.
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
o(M)
M
H*(M;\Rw)
H*(M;o(M))
For M a compact manifold, the top degree cohomology is equipped with a so-called trace morphism
\theta\colonHd(M;o(M))\to\R
Poincaré duality for differential forms is then the conjunction, for M connected, of the following two statements:
\cup\colonH*(M;\R) ⊗ Hd-*(M,o(M))\toHd(M,o(M))\simeq\R
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, i.e., a nowhere vanishing parallel section.