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Question. Is a covering space of a topological group a fiber bundle with the structure group the fundamental group of the topological space?

Let $p:E\rightarrow X$ be a covering space of X. I know how $\pi_1(X,x_0)$ acts on $p^{-1}(x_0)$. And if the space is path-wise connected all the $\pi_1(X,x_0)$ are all isomorphic for all $x_0\in X$. Thus $\pi_1(X)$ acts on the fibers.

Am I correct or am I missing something?

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Yes. In fact, any regular covering space $p\colon E\to B$ is a princpal bundle with structure group given by $\pi_1(B)/p_*(\pi_1(E))$. So if $E$ is a universal cover (which is always regular), then the structure group is just $\pi_1(B)$.

The action is given in the usual way via the monodromy action (unique lifting of paths).

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  • $\begingroup$ Can you give a reference of this ? $\endgroup$ – Babai Apr 17 '15 at 14:08
  • $\begingroup$ Any text giving an introduction to principal bundles will probably have this as an easy example. But it's also implicit in any text which describes the monodromy action of a covering on the cover. I notice it's written basically word for word on the wiki page. $\endgroup$ – Dan Rust Apr 17 '15 at 14:14
  • $\begingroup$ I understand how $\pi_1(B)$ acts on the fibers. But I don't understand how does this action factors through $\pi_1(B)/p_*(\pi_1(E))$. $\endgroup$ – Babai Apr 17 '15 at 14:23
  • $\begingroup$ @Babai The monodromy action of $\pi_1(B, x)$ on $p^{-1}(x)$ is defined by lifting a loop $\gamma$ based at $x$ to $E$ and looking at the endpoint of $\tilde{\gamma}$. $p_*(\pi_1(E)) \subset \pi_1(B)$ stabilizes $x$ as lift of any element of this subgroup is always a loop, i.e., the endpoints are the same. $\endgroup$ – Balarka Sen Apr 17 '15 at 14:33
  • $\begingroup$ Consider a simple example like the double cover $p$ of the circle $S^1$ over itself. Sure $\mathbb{Z}$ acts on the two-point fiber by either the trivial permutation if $n$ is even, or the non-trivial permutation if $n$ is odd, but we might as well mod out by the trivial permutations, which are precisely the image of $p_*$ on $\pi_1(S^1)$, the even loops. So we get a free action of $\mathbb{Z}/2\mathbb{Z}$ acting on the fiber $p^{-1}(x_0)$. $\endgroup$ – Dan Rust Apr 17 '15 at 14:34

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