- #1
sanad
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prove that if ##g:Y→Z## and ##f:X→Y## are two smooth maps between a smooth manifolds, then a homomorphism that induced are fulfilling :## (g◦f)∗=f∗◦g∗\, :\, H∙(Z)→H∙(X)##
I must to prove this by a differential forms, but I do not how I can use them .
I began in this way:
if f∗ : H(Y)→H(X), g∗ H(Z)→H(Y) , f∗ H(Y)→H(X),g∗ : H(Z)→H(Y) (by de Rham cohomology) then f∗g∗(ω): H(Z)→H(X), f∗g∗(ω): H(Z)→H(X), now I want to show that (g∘f)∗(ω)=f∗(g∗(ω)),(g∘f)∗(ω)=f∗(g∗(ω)), (g∘f)∗(ω)=ω((g∘f))(g∘f)∗(ω)=ω((g∘f)).
but I do not succeed to finish the proof.
I must to prove this by a differential forms, but I do not how I can use them .
I began in this way:
if f∗ : H(Y)→H(X), g∗ H(Z)→H(Y) , f∗ H(Y)→H(X),g∗ : H(Z)→H(Y) (by de Rham cohomology) then f∗g∗(ω): H(Z)→H(X), f∗g∗(ω): H(Z)→H(X), now I want to show that (g∘f)∗(ω)=f∗(g∗(ω)),(g∘f)∗(ω)=f∗(g∗(ω)), (g∘f)∗(ω)=ω((g∘f))(g∘f)∗(ω)=ω((g∘f)).
but I do not succeed to finish the proof.
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