Let $X$ be a path-connected space with universal cover $\widetilde{X}$, let $Y$ be another covering of $X$
$$ \widetilde{X} \hspace{1cm} \\ \searrow \\ \downarrow\hspace{.5cm} Y\\ \hspace{.25cm}\swarrow\\ X \quad\quad $$
and let $G \curvearrowright \widetilde{X}$ be a continuous group action on $\widetilde{X}$ which commutes with the action of the fundamental group by deck transformations.
Then the action descends to $Y$ in the following way: After choosing base points we know that $Y \cong \widetilde{X}/U$ for some subgroup $U$ of $\pi_1(X,x_0)$ according to the classification theorem for coverings. Now setting $$ g(Ux) = U(gx) $$ for $g \in G$ and $Ux \in Y$ we get a group action on $Y$.
Questions:
- I'm curious: Is there a more elegant description of the descended action on $Y$ that avoids the use of base points and writing $Y$ as $\widetilde{X}/U$? Maybe some uniqueness criteria or universal property? (also is it even independent of the choice of base point?)
- If the spaces are smooth manifolds and the action is smooth, will the descended action be smooth too? I would certainly think so, but I don't know how to prove it.