differential geometry w notes from teacher_Part_88

# differential geometry w notes from teacher_Part_88 - 175...

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7.2. SINGULAR CHAINS 175 2. Then, for a singular p -simplex σ p ∂∂σ p = [ σ p * ( Δ p )] = σ p * ( Δ p ) = σ * (0) = 0 ± 7.2.1 Examples Cylinder . M¨obius Band . di geom.tex; April 12, 2006; 17:59; p. 173

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176 CHAPTER 7. HOMOLOGY THEORY 7.3 Singular Homology Groups 7.3.1 Cycles, Boundaries and Homology Groups We can deﬁne the singular p -chains with coe cients in a ﬁeld K . Furthermore, we can deﬁne the multiplication of p -chains by elements of the ﬁeld K , called the scalars by, for any a , b i K , a r X i = 1 b i σ i p = r X i = 1 ab i σ i p . The chain groups C p ( M , K ) with coe cients in a ﬁeld K become inﬁnite- dimensional vector spaces. In this case the boundary homomorphism becomes a linear transformation (operator) in a vector space. Let M be a manifold and G an Abelian group. A singular
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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_88 - 175...

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