Definition of regular covering maps: independent of choice of point

Question

A covering map $p: X \to Y$ is said to be regular if for a point $x \in p^{-1}(y)$ we have $p_*(\pi_1(X,x)) \lhd \pi_1(Y,y)$.

Now my notes say that this condition is independent of the choice of the point $x \in p^{-1}(y)$ but I don't understand exactly why. Could you please explain this to me?

Thank you.

Answer 1

[I assume $X$ is path-connected, which is needed for this statement to be true.]

The subgroups $p_*(\pi_1(X,x'))$ for different choices of $x'\in p^{-1}(y)$ are exactly the conjugates of $p_*(\pi_1(X,x))$. Indeed, if you pick a path $\gamma$ from $x$ to $x'$, then you get an isomorphism from $\pi_1(X,x)$ to $\pi_1(X,x')$ by taking a loop $\alpha$ based at $x$ to the loop $\gamma^{-1}*\alpha*\gamma$ based at $x'$. Now when you apply $p_*$ to all of this, $p\circ\gamma$ is a loop, and $p_*(\pi_1(X,x'))=[p\circ\gamma]^{-1}p_*(\pi_1(X,x))[p\circ\gamma]$ as subgroups of $\pi_1(Y,y)$. (Conversely, given any $\delta\in \pi_1(Y,y)$, you can lift it to a path in $X$ from $x$ to some point $x'$ and then $\delta^{-1}p_*(\pi_1(X,x))\delta=p_*(\pi_1(X,x')$.)

In particular, if $p_*(\pi_1(X,x))$ is normal, all of its conjugates are the same, and so $p_*(\pi_1(X,x'))$ is also normal for any other $x'\in p^{-1}(y)$.

Answer 2

There is a purely algebraic theory: a morphism $p: H \to G$ of groupoids is called a covering morphism if for all objects $x$ of $H$ and element $g$ of $G$ starting at $p(x)$ there is a unique element $h$ of $H$ starting at $x$ and such that $p(h)=g$.

An element $g$ of a groupoid is called a loop if it starts and ends at the same object. The group of loops at $x$ in $G$ is written $G(x)$. The covering morphism $p$ as above is called regular if for all loops $g$ in $G$ either all or none of the elements of $p^{-1}(g)$ are loops. Again we must assume $H,G$ are connected groupoids. Then what is essentially Eric's argument as above shows that $p:H \to G$ is a regular covering morphism if and only for all objects $y$ of $H$, $p(H(y))$ is normal in $G(p(y))$.

This account is given in my book Topology and Groupoids. One advantage of this approach is that the proofs involve less notation. Another is it suggests other ideas, such as a fibration of groupoids, where one drops the uniqueness in the definition of covering morphism.