Classification of covering spaces of $\Bbb{R}\textrm{P}^2 \vee \Bbb{R}\textrm{P}^2$.

Solution 1:

This is really a long comment.

You don't need to guess how to extend the covering map. Say you have a space $X$ with universal covering $p: \tilde{X}\to X$ and you've chosen basepoints $x_0\in X$ and $e_0\in p^{-1} (x_0)$. Given a loop $a$ based at $x_0$, consider the unique lift of $a$ starting at $e_0$, and call this path $\tilde{a}$ (so $\tilde{a}(0)=e_0$).

We want to describe the covering map $f_a$ that takes $e_0$ to $\tilde{a}(1)$.

To see what $f_a$ does to some point $y\in \tilde{X}$, choose a path $\gamma$ from $y$ to $e_0$. Now I claim that $f_a (y)$ is just the endpoint of the unique lift of the composite path $p(\gamma) a \overline{p(\gamma)}$ starting at $y$. (Said differently, consider the composite path $p(\gamma) a \overline{p(\gamma)}$ in $X$, and lift it starting at $y$. The other end of this lift is $f_a (y)$.)

Note that, of course, the lift of $p(\gamma) a \overline{p(\gamma)}$ is going to start with $\gamma$.

It's a good exercise to check that this really gives a well-defined deck transformation.