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Twisted Poincaré duality

Extends Poincaré duality beyond oriented manifolds

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In mathematics, the twisted Poincaré duality is a theorem removing the restriction on Poincaré dualitytooriented 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[edit]

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 ${\displaystyle o($M)} , that is trivialized by coordinate charts of the manifold ${\displaystyle M}$, 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

${\displaystyle H^{*}(M;\mathbb {R} ^{w})}$or${\displaystyle H^{*}(M;o($M))} .

For Macompact manifold, the top degree cohomology is equipped with a so-called trace morphism

${\displaystyle \theta \colon H^{d}(M;o($M))\to \mathbb {R} } ,

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

${\displaystyle \cup \colon H^{*}(M;\mathbb {R} )\otimes H^{d-*}(M,o($M))\to H^{d}(M,o(M))\simeq \mathbb {R} }

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[edit]

• Local system
• Dualizing sheaf
• Verdier duality

References[edit]

• Some references are provided in the answers to this threadonMathOverflow.
• The online book Algebraic and geometric surgerybyAndrew 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.