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单词 HomologyOfmathbbRP3
释义

homology of 3.


We need for this problem knowledge of the homology groups of S2 and 2. We will simply assume the former:

Hk(S2;)={k=0,20else

Now, for 2, we can argue without Mayer-Vietoris. X=2 is connected, so H0(X;)=. X is non-orientable, so H2(X;) is 0. Last, H1(X;) is the abelianizationMathworldPlanetmath of the already abelianMathworldPlanetmath fundamental groupMathworldPlanetmathPlanetmath π1(X)=/2, so we have:

Hk(2;)={k=0/2k=10k2

Now that we have the homology of 2, we can compute thehomology of 3 from Mayer-Vietoris. Let X=3,V=3\\{pt}2 (by vieweing 3 as a CW-complexMathworldPlanetmath), UD3{pt}, and UVS2, where denotes equivalence through a deformation retractMathworldPlanetmath. Then the Mayer-Vietoris sequence gives

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更新时间:2025/5/4 1:47:27