Dualité de Poincaré

[invitecbade190]

Bonsoir à tous :

Voici la preuve du théorème de dualité de Poincaré ( Le "Poincaré pairing" est non dégénéré ), que je ne comprends pas bien : ( rédigé en anglais )

The proof is by induction on the number of elements in a good cover of .
If , then is diffeomorphic to , and hence, the assrtion follows from exemple ( Cet exemple se trouve dans les premiers pages de mon bouquin ).
Now, let , suppose that is a good cover, and suppos that Poincaré duality holds for every oriented - manifold with a good cover by at most open sets.
Denote by the open sets : and .
Then, the induction hypothesis asserts that Poincaré duality holds for the manifolds and .
We shall prove that satisfies Poincaré duality by considering simultaneously the Mayer - Vietoris sequences for and associated to the cover .
Thus we have commuting diagrams :



and



Commutativity of the first square ( 8.23 ) asserts that, for all closed forms and , we have :

This follows from the définition of . ( See (8.16) ).
Commutativity of the second square ( 8.23 ) asserts that, for all closed forms , ,
and we have :
This follows from the definition of . ( See ( 8.6 )
Commutativity of the diagram ( 8.24 ) asserts that, for all closed forms and , we have :
.
To see this, we recall that :
, and
Here is extended to all of by setting it equal to zero on , and
is restricted to where it still has compact support.
Since , we obtain :
as claimed.
With the commutativity of (8.23) and (8.24) established, we obtain a commuting diagram :




Since the horizontal sequences are exact and the Poincaré duality homomorphisms are isomrphisms for
, and , it follows by Five Lemma that is an isomorphism as well.
This proves the theorem.

{ \bf Questions } :
Pourriez vous m'expliquer les passages suivants :
- for all closed forms and , we have :
- for all closed forms , ,
and we have :


Le reste, je l'ai compris.
Merci d'avance.
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[invitecbade190]

Svp, pourriez vous me corriger mon code Latex qui permet d'afficher les trois diagrammes de ce topic ?
Merci d'avance.

[taladris]

Quelle livre utilises-tu? J'aurais peut-etre du temps pour t'aider mais plus tard. En attendant, tu peux consulter "Differential forms" de Bott et Tu qui explique aussi la dualite de Poincare pour les varietes.

Cordialement

[invitecbade190]

Bonjour @taladris :
D'abord, merci pour l'interet que vous portez à mon sujet.
La démonstration se trouve à la page : , du fichier pdf çi-joint.
Merci d'avance pour ton aide.

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[invitecbade190]

Un peu d'aide svp.