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differential geometry w notes from teacher_Part_94

# differential geometry w notes from teacher_Part_94 - 187...

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7.4. DE RHAM COHOMOLOGY GROUPS 187 de Rham cohomology groups are vector spaces. Let c p = r k = 1 a k σ k p be a real p -chain in M , and α be a p -form on M . We define the integral of α over c p by α, c p = c p α = r k = 1 a k σ k p α . Thus every p -form on M defines a linear functional on C p ( M ) α : C p ( M ) R , by c p α -→ α, c p . The space of all p -forms can be naturally identified with the dual space C * p ( M ) C p ( M ) C * p ( M ) . Furthermore, by Stokes theorem we have for a ( p - 1)-form d α, c p = α, ∂ c p . Thus, for every p -cycle z p , that is, if z p = 0, d α, z p = 0 , and for every closed form α , that is, if d α = 0, α, ∂ c p = 0 . More generally, let α p Z p ( M ) be a closed p -form, β p + 1 C p + 1 ( M ) be a ( p + 1)-form, z p Z p ( M ) be a p -cycle and c p + 1 C p + 1 ( M ) be a ( p + 1)-chain. Then α + d β, z p + c p = α, z p . di ff geom.tex; April 12, 2006; 17:59; p. 185

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188 CHAPTER 7. HOMOLOGY THEORY Therefore, for every equivalence class [ α p ] H p ( M ) of closed forms we can define a linear functional H p ( M ) R on the space of homology groups by, for any [ z p ] H p ( M ), [ α p ] , [ z p ] = α p , z p . This is well defined since the right hand side does not depend on the choice of representatives in the equivalence classes. This naturally identifies the space of cycles with the space of cocycles
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