Representing Covering Spaces by Permutations

Question

I am having trouble understanding the exposition in the subsection titled Representing Covering Spaces by Permutations in Section 1.3 of the book Algebraic Topology by Hatcher.

Hatcher starts by showing that given a covering map $p:(\tilde X, \tilde x_0)\to (X, x_0)$, we can define an action of $\pi_1(X, x_0)$ on $p^{-1}(x_0)$ in the following way: A member $[\gamma]$ of $\pi_1(X, x_0)$ acts on a point $\tilde x\in p^{-1}(x_0)$ and gives a point $\tilde x'$ such that the lift of $\gamma$ starting at $\tilde x'$ ends in $\tilde x$. It is clear that this indeed defines an action of $\pi_1(X, x_0)$ on $p^{-1}(x_0)$.

Then Hatcher writes:

Let $X$ be a path-connected, locally path connected, and semi-locally simply-connected space, and let $p:\tilde X\to X$ be a covering map.

Then $p:\tilde X\to X$ can be reconstructed from the associated action of $\pi_1(X, x_0)$ on $p^{-1}(x_0)$.

This is how Hathcer proceeds to show that above. Write $F=p^{-1}(x_0)$ and let $p_0:\tilde X_0\to X$ be the universal covering space for $X$. We will use the explicit construction for $\tilde X_0$, where the points in $\tilde X_0$ are homotopy classes of paths starting at $x_0$. Define a map $h:\tilde X_0\times F\to \tilde X$ as $h([\gamma], \tilde x_0)=\tilde \gamma(1)$, where $\tilde \gamma$ is the lift of $\gamma$ in $\tilde X$ starting at the point $\tilde x_0$. Then we note that $h$ is a surjective local homeomorphism. Thus $\tilde X$ is homeomorphic to the quotient of $\tilde X_0\times F$ determined by the equivalence relation induced by $h$.

Then Hatcher shows that the $h$-fibre of a point $([\gamma], \tilde x_0)$ in $\tilde X_0\times F$ is all the points of the form $([\lambda\gamma], L_\gamma(\tilde x_0))$. Here $\lambda$ is any loop based at $x_0$ in $X$ and $L_\lambda(\tilde x_0)$ denotes the point $\tilde x_0'$ in $F$ such that the lift of starting at $\tilde x_0'$ ends at $\tilde x_0$.

Let us denote the quotient space by $\tilde X_\rho$, where $\rho$ is the homomorphism $\pi_1(X, x_0)\to \text{Perm}_F$.

I understand everything so far. But then Hacther writes the following which I do not understand:

Note that the definition of $\tilde X_\rho$ makes sense whenever we are given an action $\rho$ of $\pi_1(X,x_0)$ on a set $F$. There is a natural projection $\tilde X_\rho\to X$ sending $([\gamma], \tilde x_0)$ to $\gamma(1)$, and this is a covering space since if $U\subseteq X$ is an open set over which the universal cover $\tilde X_0$ is a product $U\times \pi_1(X, x_0)$, then the identifications defining $\tilde X_\rho$ simply collapse $U\times \pi_1(X, x_0)\times F$ to $U\times F$.

Here are the things not clear to me:

1) How is $\tilde X_\rho$ defined given an action of $\pi_1(X, x_0)$ on a given set $F$. Here we do not have the covering map $p:\tilde X\to X$. My guess is that it is defined as follows: $\tilde X_\rho$ is the quotient of $\tilde X_0\times F$ by the equivalence relation defined as $([\gamma], \tilde x_0)\sim ([\lambda\gamma], [\lambda]\cdot\tilde x_0)$, where $[\lambda]\in \pi_1(X, x_0)$.

2) If my guess is correct, then the map $\tilde X_\rho\to X$ sending $([\gamma], \tilde x_0)$ to $\gamma(1)$ makes no sense to me. For $([\gamma], \tilde x_0)$ is not even a member of $\tilde X_\rho$.

3) I see that $p_0^{-1}(U)$ can be thought of as $U\times \pi_1(X, x_0)$ where $U$ is evenly covered by $p_0$. This is because the number of sheets of $\tilde X_0$ is in bijective correspondence with $\pi_1(X, x_0)$. But I do not understand what Hatcher means by "the identifications defining $\tilde X_\rho$ simply collapse $U\times \pi_1(X, x_0)\times F$ to $U\times F$."

4) My main concern is to get a a covering map $\tilde X_\rho\to X$ such that $\bar h:\tilde X_\rho\to \tilde X$ becomes an isomorphism of covering spaces, and I do not see how to do this.

Thank you.

Answer 1

1) You are correct. In the case that there is a covering map $p:\widetilde{X} \to X$, $\widetilde{X}_\rho$ is defined to be the quotient of $\widetilde{X}_0\times F$ by identifying all pairs $([\gamma],\widetilde{x}_0)$ and $([\lambda \cdot \gamma],L_\lambda(\widetilde{x}_0))$ for all $[\lambda] \in \pi_1(X,x_0)$. The only property of the covering space used in this definition is the action of $\pi_1(X,x_0)$ on $p^{-1}(x_0)$ determined by the covering space. In this case the action of $\pi_1(X,x_0)$ is given by the homomorphism $[\gamma] \mapsto L_\gamma$. If instead you are just given an action $\rho$ of $\pi_1(X,x_0)$ on some set $F$, then $\rho$ takes the place of $L_\gamma$.

2) The element $([\gamma],\widetilde{x}_0)$ is a representative of an element in the quotient space $\widetilde{X}_\rho$. Certainly $[\gamma]$ is an element of $\widetilde{X}_0$, and $\widetilde{x}_0$ is just some element of $F$.

3) The claim is that $U\times [\lambda] \times F$ gets identified with $U \times [\lambda']\times F$ in the quotient space $\widetilde{X}_\rho$. What do $U\times [\lambda]$ and $U \times [\lambda']$ really mean? Well a point in $U \times[\lambda]$ is just a point $[\gamma] \in \widetilde{X}_0$. Moving from $U \times [\lambda]$ to $U \times [\lambda']$ corresponds to moving $\rho([\lambda])\cdot\widetilde{x}_0$ to $\rho([\lambda'])\cdot\widetilde{x}_0$, which is accomplished by applying $\rho([\lambda'\cdot \bar{\lambda}])$, as $\rho([\lambda' \cdot \bar{\lambda}])\cdot \rho([\lambda])\cdot\widetilde{x}_0=\rho([\lambda'\cdot \bar{\lambda}\cdot \lambda])\cdot\widetilde{x}_0=\rho([\lambda'])\cdot\widetilde{x}_0$. So the point $[\gamma] \in \widetilde{X}_0$ corresponds to the point $[\lambda'\cdot \lambda\cdot \gamma]$ in $U \times [\lambda']$. In the quotient $\widetilde{X}_\rho$ we identify $([\gamma],\rho([\lambda])\cdot \widetilde{x}_0)$ with $([\lambda'\cdot \bar{\lambda}\cdot \gamma],\rho([\lambda'\cdot \bar{\lambda}])\cdot \rho([\lambda])\cdot \widetilde{x}_0)$. So $U\times[\lambda]\times F$ is identified with $U \times [\lambda']\times F$ (as $\rho$ is a group action, so varying $\widetilde{x}_0$ to range over all points in $F$ means both $\rho([\lambda])\cdot \widetilde{x}_0$ and $\rho([\lambda'\cdot \bar{\lambda}])\cdot \rho([\lambda])\cdot \widetilde{x}_0$ will range over all points in $F$).

4) The map $h$ defined by Hatcher (the "surjective local homeomorphism") yields a well-defined map $\bar{h}$ on the quotient space because $([\gamma],\widetilde{x}_0)\sim([\gamma'],\widetilde{x}_0')$ implies $h([\gamma],\widetilde{x}_0)=h([\gamma'],\widetilde{x}_0)$. The converse holds as well, which implies that $\bar{h}$ is injective on the quotient space. So $\bar{h}$ is the isomorphism of covering spaces you seek. It doesn't make sense to look for such a map in the case where you're only given the action $\rho$ and not a covering space $\widetilde{X}\to X$, because after all you are trying to define a map to the covering space $\widetilde{X}$.