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Let $p:E\to X$ be a covering space and $\pi_1(E)$ be a fundamental group of $E$. Can you give me a recept for calculating a fundamental group $\pi_1(X)$ (may be for some special cases)?

Thanks a lot!

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    In the situation you stated, I don't think we can say much more than that $\pi_1(E)$ is isomorphic to a subgroup of $\pi_1(X)$. (To compute $\pi_1(X)$, I believe you usually have to refer to the universal cover of $X$ and prove something manually, or use Seifert-van Kampen theorem and known fundamental groups.)2012-10-16
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    Just to echo @Tunococ, there's a covering space for every conjugacy class of subgroups of $\pi_1(X)$, and one approach to computing $\pi_1(X)$ once you've found $E$ to be the universal cover is as the set $p^{-1}(x_0)$ with the group structure of the deck transformations on $E$.2012-10-16

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