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{{Technical|date=July 2011}}
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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==
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>.
 
==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 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
:<math>H^* (M; \mathbb R^w)</math> or <math>H^* (M; o(M))</math>.
 
For ''M'' a ''compact''  manifold, the top degree cohomology is equipped with a so-called '''trace morphism'''
:<math>\theta: H^d (M; o(M)) \to \mathbb R</math>,
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
:<math>\cup : H^* (M; \mathbb R)\otimes H^{d-*}(M, o(M)) \to H^d(M, o(M)) \simeq \mathbb R</math>
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==
*[[Local system]]
*[[Dualizing sheaf]]
*[[Verdier duality]]
 
==References==
*Some references are provided in [http://mathoverflow.net/questions/61194/non-oriented-version-of-poincare-duality the answers to this thread] on [[MathOverflow]]
*The online book [http://www.maths.ed.ac.uk/~aar/books/surgery.pdf ''Algebraic and geometric surgery''] by Andrew Ranicki
*[[Raoul Bott|R. Bott]]-L. Tu. ''Differential forms in algebraic topology'', a classic reference
 
{{DEFAULTSORT:Twisted Poincare duality}}
[[Category:Algebraic topology]]
[[Category:Manifolds]]
[[Category:Duality theories]]
[[Category:Theorems in topology]]

Latest revision as of 11:16, 2 January 2015

When you pledge any really love and also produce your personal future girlfriend employing the mark of one's life jointly promote the lady an exceptional gemstone under a strong intending! A difficult diamond gemstone would be that recognizable as being the thank you give out and also the best declaration of one's personal method! Rejoice your personal future lifetime together with this important token of one's really love.Precisely better method to state I detest this one Ring than to whether obliterate or maybe lose the product. There is put together a list of one particular suspicious and also frequent solutions to insuring the best lost or maybe busted band. OK best of luck! Since when you are planning on your proposal you could end fantasizing on the jewelry on top! Still hello no less than youll feel lowering costs during an excellent function additionally a pretty dress and. Your entire day should really be costly certainly not reasons why run moved broke! Obviously you can help out faith slightly with hinting towards dude type of call youd just like providing you do not disregard those funds isnt it all! You may need it for several stuff still do not see it since primary factor within your nuptials thereby wedding ring! Make certain it's ready and trouble-free if you'll still like to pick your 5 million rock best of luck choosing personally your Jay-Z.One in the passionate things that might happen for a girl will be wanted from the gentleman he/she loves to marry him or her! There are lots of steps some guy could possibly propose but there is an obvious thing which is going to often be generally there the moment they popular the question a wedding ring.this really is any double whammy..! A health club is among any no go zones concerning involvement bands..! Through slamming to banging ones band on heavy weights to opportunity seekers entering lockers: there is obscurity concerning any ring to cover up over a health club..! If you need to ensure any lost band: bring it available!If you haven't become the purpose but just don't bring it to a health club in the first place!!! Actually much easier that carefully keep ones gemstone using a secure place at your house as opposed to to face the loss of specific.2.The average whole diamonds carat body weight utilized in wedding jewelry try projected around .37 carats around 1 completely carat.A rough gemstone may whole gem stone obtained in type..! Around vast amounts of a very long time regarding wanted gemstones happened to be structured for the globe..! Initial noticed in Asia in 700 B.C., tough expensive diamonds happened to be thanks powers which got thought that changing all of them might reduce their power..! Imagine the scary provided early people were decide some sort of reduce and also refined stone wedding rings presently.This circumstance is really prevalent their occurances commonly always recorded and yet relax knowing this occurs each day. Rings are left out only reserved for a short while and neglected or possibly adjust in one counter just to evaporate instant later on to never be seen anymore.Standing outside of the crowd may desire your some getting ready to have married! A fairly easy action that will permit just about any some to do this target looks beautifying exclusive accessories! Harry Winstons involvement bands allows one prospective some toward supersede her or his expectations within the thing! Your jewelry are a couple of the best accepted involvement bands on every favored jewel search! All these bands aren't only popular among the known folks prominent celebs presents Winstons higher any other forms of jewelry! By way of example, celeb Ben Aflleck, obtained one six carat gemstone Winston engagement ring over Jennifer Lopez! Ben Affleck plus Jennifer Lopez, however, are not the only celebrities shown to wear the trendy jewelry! Richard switch, one other favored celeb, could have purchased one 65 carat diamond ring over Elizabeth Taylor.Like much of the cases with list: the simple answer is to simply not necessarily own your own Engagement Ring if you are deciding upon gardening. Room your own ring-in a safe place before you start this type of weeding!There was one such news background around lady who states this girl never used their bands health and fitness club..! Eco-friendly diamonds But one day this girl was managing later part of the to your job to assumed that she could possibly basically depart their wedding rings in her own locker right at the fitness center..! Sure-enough, their bands, treasured on 16,000 had been eliminated soon.Also their reduce on a large gemstone is a very important practice choose that cut that will enhance the great thing about that 4 carat gemstone. Them ring wouldn't be discounted, the ones that could afford these are the ones nasty high men and women and others that have conserve a large budget for the company's gemstone.This hints and tips can be employed two practices 1st for people who are even over to kill or alternatively lose those ring, and also second for folks who like to submit a good number of excess methods to ensure the security and safety of the many loved control.

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