# Covering space and Fundamental group

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.) – Tunococ Oct 16 '12 at 8:47
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 $E$. – Kevin Carlson Oct 16 '12 at 9:34

You cannot talk in such a generality.... In general, calculating the fundamental group is a "difficult" problem. There are no algoritms such as you were searching for.

Related with coverings, there are a lot of theorems that help you to calculate in special cases. $\pi _ 1 (B, b)/ p _\ast (\pi _ 1(E,e))$ is a $G$-set isomorphic to the fiber and so on.

But there are a lot of nice special cases. I will give a nice one - and I hope you enjoy as I enjoyed when I figured it out for the first time.

If $G$ is a topological group, so is any covering space of $G$. This is an exercise of liftings: in this exercise, you should prove also that the covering is itself a homomorphism of topological groups... But, if $p: E\to G$ is a covering, you get a nice understanding of the fundamental group of $G$.

First of all, you know this is a regular cover - since the fundamental group of any topological group is abelian (this is an exercise of Eckman-Hilton clock).

Second, it is easy to prove that the group of the automorphisms of $p$ is isomorphic to the Kernel of $p$. So, you got that $\pi _ 1(G,g)/p _ \ast (\pi _1 (E,e))$ is isomorphic to the kernel of the covering.

Look at the particular case of universal coverings - you get that the kernel is the fundamental group of the base. It is the case of the universal covering of $S^1$.

If you don't think it's nice enough, you should investigate more the particular case of topological groups and get more nice results... And you can specialize your question and get more interesting results.

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