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Is there any one can tell me the relationship between the fundamental group and the cohomology group H1(X).

I am appreciate with your help.

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 Please check the FAQ's suggestions of other sites which might be more appropriate. – Douglas Zare Aug 5 at 10:06
I think he means the homology group $H_1(X)$. It was known near the beginning of the 20th century that for $X$ connected, $H_1(X)$ is $\pi_1(X,x)$ made abelian. This suggested a search for higher dimensional generalisations of the fundamental group, but these were not found till the 1970s as generalisations of the fundamental groupoid $\pi_1(X,A)$ on a set $A$ of base points. See my paper arXiv:1012.2824 "Covering morphisms of groupoids, derived modules and a 1-dimensional Relative Hurewicz Theorem " for more background on the 1-dimensional case. So the question is suggestive. – Ronnie Brown Aug 6 at 9:54

closed as off topic by Douglas Zare, Dan Petersen, Misha, David Roberts, Mark Grant Aug 5 at 11:30

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