differential geometry w notes from teacher_Part_87

# differential geometry w notes from teacher_Part_87 - 173...

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7.2. SINGULAR CHAINS 173 Let σ p : Δ p M be a singular p -simplex in M . Then its boundary ∂σ p is the integer ( p - 1)-chain in M deﬁned by ∂σ p = σ p * ( Δ p ) . In more detail ∂σ p = σ p * ( Δ p ) = p X k = 0 ( - 1) k σ p * ( Δ ( k ) p - 1 ) = p X k = 0 ( - 1) k ( σ p f k ) . Recall that σ p * ( Δ ( k ) p - 1 ) = ( σ p f k ) is the k -th face of the singular p -simplex σ p . That is, the boundary of the image of Δ p is the image of the boundary of Δ p . The boundary of any singular p -chain with coe cients in G is deﬁned by, for any g k G , σ k p S p ( M ), r X k = 1 g k σ k p = r X k = 1 g k ∂σ k p . This leads to the boundary homomorphism : C p ( M ; G ) C p - 1 ( M ; G ) . Let F : M V be a map, σ p be a singular p -simplex in M and F * σ p be the induced singular p -simplex in V . Then ( F * σ p ) = ( F σ p ) = ( F σ p ) * ( Δ p ) = ( F * σ p * )( Δ p ) = F * [ σ p * ( Δ p )] = F * ( ∂σ p ) More generally, let c p = r X k = 1 g k σ k p be a p -chain on M and F * c p be the induced p -chain on V . di geom.tex; April 12, 2006; 17:59; p. 171

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174 CHAPTER 7. HOMOLOGY THEORY Then ( F * c p ) = F * ( c p ) . Therefore, F * = F * ∂ , in other words, the boundary of an image is the image of the boundary . Thus, we obtain a
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## This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

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differential geometry w notes from teacher_Part_87 - 173...

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