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.
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Let M be a d-dimensional compact boundaryless differential manifold with orientation character w(M). Then the cap product with the w- twisted fundamental class induces Poincaré duality isomorphisms between homology and cohomology: and .
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 , that is trivialized by coordinate charts of the manifold NM, with transition maps the sign of the Jacobian determinant of the charts transition maps. 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, 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.