# Twisted Poincaré duality

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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 , 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:

- The trace morphism is a linear isomorphism,
- The cup product, or exterior product of differential forms

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.

## See also

## References

- Some references are provided in the answers to this thread on MathOverflow
- The online book
*Algebraic and geometric surgery*by Andrew Ranicki - R. Bott-L. Tu.
*Differential forms in algebraic topology*, a classic reference