differential geometry w notes from teacher_Part_86

differential geometry w notes from teacher_Part_86 - 7.2....

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Unformatted text preview: 7.2. SINGULAR CHAINS 171 • Examples. • The boundary of a standard simplex is not a simplex, but an integer ( p- 1)- chain . • Let G be an Abelian group. A singular p-chain on M with coe ffi cients in G is a finite formal sum c p = r X k = 1 g k σ k p of singular simplexes σ k p : Δ p → M with coe ffi cients g k which are elements of the group G . • Examples. • Let S p ( M ) be the set of all singular p-simplexes in M . Then, a p-chain is a function c p : S p ( M ) → G , such that its value is not equal to zero only for finitely many simplexes . These simplexes are exactly σ k p listed in the formal sum, and the values of the function c p are exactly the coe ffi cients of the formal sum, that is c p ( σ k p ) = g k . • Alternatively, a p-chain can be thought of as a finite subset of S p ( M ) × G , that is, a finite set of ordered pairs c p = { ( σ k p , g k ) } r k = 1 • The notation of a p-chain as a sum, or as a function, is useful since we can define the addition of...
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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_86 - 7.2....

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