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Unformatted text preview: 7.3. SINGULAR HOMOLOGY GROUPS 177 • The set of all pboundaries 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 pboundary group . • Obviously the pboundary group is the image of the boundary homomor phism B p ( M ; G ) = Im ∂ p + 1 . • Since every pboundary is a pcycle (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 pcycles are homologous if they di ff er by a boundary. • The set of equivalence classes of pcycles homologous to each other, that is, the quotient group H p ( M ; G ) = Z p ( M ; G ) / B p ( M ; G ) , is called the pth homology group ....
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 Spring '10
 Wong
 Geometry, Algebraic Topology, singular homology groups, homology groups, compact manifold

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