Deck transformations of universal cover are isomorphic to the fundamental group - explicitly

Question

I have read on several places that given a (say path connected) topological space $X$ and its universal covering $\tilde{X}\stackrel{p}\rightarrow X$, there is an isomorphism

$$\mathrm{Deck}(\tilde{X}/X) \simeq \pi_1(X, x_0).$$

Here $\mathrm{Deck}(\tilde{X}/X)$ denotes the group of deck transformations of the universal cover and $x_0$ is some basepoint. (Maybe I am omitting some assumptions, but I am interested in this mainly for path connected smooth manifolds, which should be "nice enough" topological spaces.)

However, I was unable to find any explicit description of the isomorphism. What I have in mind is the following:

Given a loop $\gamma$ in $X$, what is the corresponding deck transformation? In other words, how does the fundamental group $\pi_1(X, x_0)$ act on the covering space $\tilde{X}$?

That is, I am interested in the "geometric picture" behind the isomorphism. I understand there is some choice involved, something like fixing some preimage $\tilde{x}_0 \in p^{-1}(x_0)$, I can imagine lifting the loop uniquely modulo this choice, however I cannot see how to obtain a homemorphism of $\tilde{X}$ using this lifted path (is it some use of the universal property of $\tilde{X}\rightarrow X$, perhaps?).

Thanks in advance for any help.

Answer 1

Here is how it goes.

Let $B$, be a space nice enough to have a (simply connected) universal cover, say $B$ is connected, locally connected and semi-locally simply connected. Let $(X,x_0)\to (B,b_0)$ be its universal cover.

Take a loop $\gamma: (S^1,1)\to (B,b_0)$ then you can lift $\gamma$ to a path $\overline{\gamma}: I\to X$ that projects to $\gamma$. Now $\overline{\gamma}(1)$ is an element of $X_{b_0}$. You can use then the following theorem.

Let $(Y,y_0)\to (B,b_0)$ be a (path) connected and locally path connected space over $B$ and $(X,x_0)\to (B,b_0)$ is a cover of $B$, then a lift of $(Y,y_0)\to (B,b_0)$ to $(Y,y_0)\to (X,x_0)$ exists iff the image of $\pi_1(Y,y_0)$ inside $\pi_1(B,b_0)$ is contained in the image of $\pi_1(X,x_0)$ inside $\pi_1(B,b_0)$

Use the previous theorem with $(Y,y_0)=(X,\overline{\gamma}(1) )$. This tells you that there exists a covering map $X\to X$ sending $x_0$ to $\gamma(1)$.

It is easy to see that this map depends only on the homotopy class of $\gamma$ using the following result

Let $(X,x_0)$ be a cover of $(B,b_0)$ and $Y$ be a connected space over $B$. If two liftings of $Y\to B$ to $Y\to X$ coincide at some $y_0$ in $Y$, then they're equal.

This tells you that if $\overline{\gamma}(1)=\overline{\tau}(1)$ then the two morphisms $X\to X$ you get, coincide. Moreover, using the inverse of $\gamma$, you see that the morphisms $X\to X$ you get are automorphisms.

This gives you a well defined map $\pi_1(B,b_0)\to \text{Aut}_B(X)$. Using what I said before, it is easy to see that it is an isomorphism.

Answer 2

I felt there was still a bit more to say about a "geometric picture" behind this action, so here it goes.

Right, so take $X$ be a suitably nice space with simply connected cover $p:\tilde{X}\rightarrow X$. Take $\gamma$ to be a loop in $X$ based at some point $x_0\in X$, representing an element of $\pi_1(X,x_0)$. Now, pick some $\tilde{x}\in\tilde{X}$; we'd like to describe $\gamma\cdot\tilde{x}$, the image of $\tilde{x}$ under the deck transformation determined by $\gamma$.

Let $\tilde{x_0}$ be some lift of $x_0$ to $X$ and pick a path $\tilde{\alpha}$ in $\tilde{X}$ starting at $\tilde{x_0}$ and ending at $\tilde{x}$. Define $\alpha=p\circ\tilde{\alpha}$. Let $\gamma\bullet\alpha$ be the path concatenation of $\gamma$ followed by $\alpha$ and let $\widetilde{\gamma\bullet\alpha}$ be the lift of $\gamma\bullet\alpha$ to $\tilde{X}$ starting at $\tilde{x_0}$. Then, define $\gamma\cdot\tilde{x}=\widetilde{\gamma\bullet\alpha}(1)$. The facts stated in another answer above tell us that this is well defined.

This way to define the action yields a nice picture, I think. The idea is that if $\tilde{\gamma}$ is the lift of $\gamma$ starting at $\tilde{x_0}$, then $\tilde{\gamma}(1)$ is another element of $p^{-1}(x_0)$. Then, we can "do $\tilde{\alpha}$" starting at $\tilde{\gamma}(1)$ instead of $\tilde{x_0}$, and $\gamma\cdot\tilde{x}$ is the endpoint of this new path. See the image below.