differential geometry w notes from teacher_Part_90

differential geometry w notes from teacher_Part_90 - 179...

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7.3. SINGULAR HOMOLOGY GROUPS 179 Let M and V be manifolds, G be a n Abelain group, F : M V be a map and F * : C p ( M ; G ) C p ( V ; G ) be the induced homomorphism of chain groups. Since the induced homomorphism F * commutes with the boundary homo- morphism , the groups Z p ( M ; G ) and B p ( M ; G ) are closed under F * . Therefore, the homomorphism F * naturally acts on the homology groups F * : H p ( M ; G ) H p ( V ; G ) . If F : M V is a homeomorphism, then there is the inverse homeomor- phism F - 1 : V M and the inverse induced homomorphism F - 1 * : H p ( V ; G ) H p ( M ; G ) . In this case, the induced homomorphism F * is an isomorphism. Theorem 7.3.2 Let M and V be compact homeomorphic manifolds and G be an Abelian group. Then their homology groups are isomor- phic, that is, for any p H p ( M ; K ) H p ( V ; G ) . Proof : Follows from above. Thus, homology groups are topological invariants . Corollary 7.3.2 Let M and V be compact manifolds. If there is an Abelian group G and an integer p such that their homology groups are
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