Example of topology

Question

In my notes I read that, given a covering map $p:E\to X$, $E$ is called a $G$-covering if $G=\operatorname{Aut}(E,p)$ and $X=E/G$. Can you make an example of a covering that is not a $G$-covering? Thank you in advance

Answer 1

Here is the simplest example I am aware of: consider the wedge of two circles $X := S^1\vee S^1$. The basepoint $p$ will be the basepoint of the wedge. Label one of the circles $a$ and the other $b$, and give them an orientation. The cover $E$ will have three lifts of the basepoint, call them $x,y,z$, and six $1$-cells. A label and choice of orientation on the $1$-cells defines a covering map $E \to X$.

Let $E$ have three $1$-cells attached to $x$: one, labeled $a$ is a loop, the other two are labeled $b$ and given opposing orientations and are attached with one side to $x$ and the other to $y$. There will be two more $1$-cells attaching to $y$, each attaching on the other side to $z$. Label these $1$-cells $a$ and give them opposing orientations. Finally, attach a $1$-cell labeled $b$ to $z$ (on both sides). (Draw the picture!!)

It's easy to see that this defines a cover $E \to X$, and that $\operatorname{Aut}(E,p)$ is trivial, so $E$ is not a $G$-cover.