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| {{Technical|date=July 2011}}
| | 28 year old Medical Oncologist Standifer from Bloomfield, spends time with hobbies including jetski, [http://saintsofgodchurch.com/?p=76 yes chef] hack and antiques. Loves to visit unknown towns and spots like Portovenere. |
| In mathematics, the '''twisted Poincaré duality''' is a theorem removing the restriction on [[Poincaré duality]] to [[oriented manifold]]s. The existence of a global orientation is replaced by carrying along local information, by means of a [[local coefficient system]].
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| ==Integer-valued formulation==
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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: <math>H^* (M) \to H_{d - *}(M;\mathbb Z^w)</math> and <math>H^* (M; \mathbb Z^w) \to H_{d - *}(M)</math>.
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| ==Twisted Poincaré duality for de Rham cohomology==
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| Another version of the theorem with real coefficients features the [[de Rham cohomology]] with values in the '''orientation bundle'''. This is the '''[[flat vector bundle|flat]]''' real [[line bundle]] denoted <math> o(M)</math>, 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 vector bundle|flat line bundle]], it has a de Rham cohomology, denoted by
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| :<math>H^* (M; \mathbb R^w)</math> or <math>H^* (M; o(M))</math>. | |
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| For ''M'' a ''compact'' manifold, the top degree cohomology is equipped with a so-called '''trace morphism'''
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| :<math>\theta: H^d (M; o(M)) \to \mathbb R</math>,
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| that is to be interpreted as integration on ''M'', ''ie.'' evaluating against the fundamental class.
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| The Poincaré duality for differential forms is then the conjunction, for ''M'' connected, of the following two statements:
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| * The trace morphism is a linear isomorphism,
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| * The cup product, or [[exterior product]] of differential forms
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| :<math>\cup : H^* (M; \mathbb R)\otimes H^{d-*}(M, o(M)) \to H^d(M, o(M)) \simeq \mathbb R</math>
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| is non-degenerate.
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| 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.
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| ==See also== | |
| *[[Local system]]
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| *[[Dualizing sheaf]]
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| *[[Verdier duality]]
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| ==References==
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| *Some references are provided in [http://mathoverflow.net/questions/61194/non-oriented-version-of-poincare-duality the answers to this thread] on [[MathOverflow]]
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| *The online book [http://www.maths.ed.ac.uk/~aar/books/surgery.pdf ''Algebraic and geometric surgery''] by Andrew Ranicki
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| *[[Raoul Bott|R. Bott]]-L. Tu. ''Differential forms in algebraic topology'', a classic reference
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| {{DEFAULTSORT:Twisted Poincare duality}}
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| [[Category:Algebraic topology]]
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| [[Category:Manifolds]]
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| [[Category:Duality theories]]
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| [[Category:Theorems in topology]]
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28 year old Medical Oncologist Standifer from Bloomfield, spends time with hobbies including jetski, yes chef hack and antiques. Loves to visit unknown towns and spots like Portovenere.