I am considering the following problem: $\tilde X$ path connected, $X$ path connected, locally path connected, $P:\tilde X \to X$ covering map, $x_0 \in X, \tilde x_0, \tilde x_1 \in p^{-1}(x_0).$ I would like to find a necessary and sufficient condition for the existence of a deck transformation sending $\tilde x_1$ to $ \tilde x_0$.
My lecture notes say that 1) $p_\star(\pi_1(\tilde X, \tilde x_1)) \subset p_\star(\pi_1(\tilde X, \tilde x_0))$ is necessary and sufficient. I know that 1) gives us a unique lift of p, say $\tilde p$, mapping $\tilde x_1$ to $\tilde x_0$ but I don't see why this has to be a homeomorphism?
In my opinion need the opposite inclusion of 1) as well to get a lift $\tilde p_1$ sending $\tilde x_0$ to $\tilde x_1$, and can then deduce by the uniqueness of lifts that $\tilde p \circ \tilde p_1 \cong id_{\tilde X} $ and thus $\tilde p$ is a homemorphism. Can anyone help me this this? Thanks.
You are right. It should be $p_\star(\pi_1(\tilde X, \tilde x_1)) = p_\star(\pi_1(\tilde X, \tilde x_0))$. You can find it in Hatcher's book.