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Unformatted text preview: SYMPLECTIC GEOMETRY, LECTURE 2 Prof. Denis Auroux 1. Homology and Cohomology Recall from last time that, for M a smooth manifold, we produced a graded differential algebra (Ω ∗ ( M ) , ∧ , d ) giving us a cohomology H ∗ ( M ) with cup product [ α ] ∪ [ β ] = [ α ∧ β ] (which is welldefined since d ( α ∧ β ) = dα ∧ β + ( − 1) deg α α ∧ dβ and ( α + dη ) ∧ β = α ∧ β + dη ∧ β ). Furthermore, we obtain a pairing with homology: for Σ ⊂ M a pdimensional, oriented, closed submanifold with associated class [Σ] ∈ H p ( M ), we define (1) [ α ] , [Σ] = α Σ for [ α ] ∈ H p ( M, R ), and extend this by linearity to give a pairing with all of H p ( M ). That this is welldefined is a consequence of Stokes’ theorem: (2) dα = α Σ ∂ Σ Remark. A form is closed its integral on submanifolds depends only the homology class of the submanifold. ⇔ Furthermore, if M n is compact, closed, and oriented, we have a nondegenerate pairing (3) H p ( M, R ) ⊗ H n − p ( M, R ) → R , [ α ] ⊗ [ β ] → M α ∧ β which induces the Poincar´ e duality H n − p ( M, R ) H p ( M, R ). In the noncompact case, we have the same → statement using cohomology with compact support H C n − p ( M, R ).)....
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This note was uploaded on 03/08/2010 for the course MATHEMATIC 570 taught by Professor Denisauroux during the Spring '10 term at Caltech.
 Spring '10
 DenisAuroux
 Algebra, Geometry

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