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Twisted Poincaré duality

From Wikipedia, the free encyclopedia

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

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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 , that is trivialized by coordinate charts of the manifold , 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

or .

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:

  • 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, i.e., a nowhere vanishing parallel section.

See also

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References

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  • Some references are provided in the answers to this thread on MathOverflow.
  • The online book Algebraic and geometric surgery by Andrew Ranicki.
  • Bott, Raoul; Tu, Loring W. (1982). Differential forms in algebraic topology. Graduate Texts in Mathematics. Vol. 82. New York-Berlin: Springer-Verlag. doi:10.1007/978-1-4757-3951-0. ISBN 0-387-90613-4. MR 0658304.