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单词 MayerVietorisSequence
释义

Mayer-Vietoris sequence


Let X is a topological spaceMathworldPlanetmath, and A,BX are such that X=int(A)int(B), and C=AB. Then there is an exact sequencePlanetmathPlanetmathPlanetmath of homology groups:

Hn(C)i*-j*Hn(A)Hn(B)j*+i*Hn(X)*Hn-1(C)

Here, i* is induced by the inclusions i:BX and j* by j:AX, and * is the following map:if x is in Hn(X), then it can be written as the sum of a chain in A and one in B, x=a+b.a=-b, since x=0. Thus, a is a chain in C, and so representsa class in Hn-1(C). This is *x. One can easily check (by standard diagram chasing) that this map is well defined on the level of homologyMathworldPlanetmathPlanetmath.

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更新时间:2025/5/3 22:01:14