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$E, F, B$ are topological space, $B$ path connected. If we have given a long exact sequence.. $$\cdots\to \pi_1(F)\to \pi_1(E)\to \pi_1(B)\to\cdots$$

what will the relationship of $H_1(F,\mathbb R)$, $H_1(E,\mathbb R)$ and $H_1(B,\mathbb R)$

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First of all, what of kind of arrow do you have? Embeddings? Second of all. Secondly to define $\pi_1(X)$ X has to be locally path-connected and connected. Moreover, to define $H_1(X,\mathbb R)$ you need more in general: a simplicial structure, or manifold, etc. You must be more precise, specially with arrows. –  Ilies Zidane Aug 29 '12 at 20:04
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Why should we need any more structure than topological on $X$ to define, say, the singular homology? Also, this looks like a long exact sequence to me. Should there be zeroes on the ends? –  Kevin Carlson Aug 29 '12 at 20:22
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Have you seen en.wikipedia.org/wiki/Serre_spectral_sequence? Your notation suggests (assuming you mean "long exact sequence" rather than "short exact sequence") that $F \to E \to B$ is a fiber sequence, in which case the Serre spectral sequence lets you describe the homology of $E$ in terms of that of $F$ and $B$. –  Akhil Mathew Aug 29 '12 at 20:22
    
Why should we need any more structure than topological on $X$ to define $\pi_1(X)$? –  a.r. Aug 29 '12 at 21:25
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