differential geometry w notes from teacher_Part_89

differential geometry w notes from teacher_Part_89 - 7.3...

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Unformatted text preview: 7.3. SINGULAR HOMOLOGY GROUPS 177 • The set of all p-boundaries in M B p ( M ; G ) = { b p ∈ C p ( M ; G ) | b p = ∂ c p + 1 for some c p + 1 ∈ C p + 1 ( M ; G ) } is a subgroup of the chain group C p ( M ; G ) called the p-boundary group . • Obviously the p-boundary group is the image of the boundary homomor- phism B p ( M ; G ) = Im ∂ p + 1 . • Since every p-boundary is a p-cycle (because of ∂ 2 = 0) the group B p ( M ; G ) is a subgroup of Z p ( M ; G ). • In the case, when G = K is a field, then B p ( M ; K ) is a vector subspace of the vector space Z p ( M ; K ). • Let M be a manifold and G be an Abelain group. We say that two p-cycles are homologous if they di ff er by a boundary. • The set of equivalence classes of p-cycles homologous to each other, that is, the quotient group H p ( M ; G ) = Z p ( M ; G ) / B p ( M ; G ) , is called the p-th homology group ....
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differential geometry w notes from teacher_Part_89 - 7.3...

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