Lift of $z^2$ map $S^1 \to S^1$

Question

$\newcommand{\id}{\operatorname{id}}$It can be proved that identity map $\id: S^1 \to S^1$ does not lift to $\widetilde{\id} : S^1 \to \mathbb{R}$ such that $e^{\widetilde{\id(z)}i} = z$.

The standard proof is based on injectivity contradiction (the lift can not be bijective because otherwise it will be a homeomorphism and it is not injective than for some $z_0, z_1 \in S^1$ we have $z_0 = e^{\widetilde{\id(z_0)}i} = e^{\widetilde{\id(z_1)}i} = z_1$. Contradiction).

It seems like this contraction doesn't work for, say, $z \mapsto z^2$. So does this map lift to $S^1 \to \mathbb{R}$?

Answer 1

In a word, "no".

If $\tilde{f}:S^{1} \to \mathbf{R}$ is an arbitrary continuous map, then its image is a compact, connected set of real numbers, i.e., a point or a closed, bounded interval. In any case, $\tilde{f}$ achieves an absolute maximum at some point $z_{0}$ of $S^{1}$. If $\pi:\mathbf{R} \to S^{1}$ is the covering map $\pi(t) = e^{it}$, then $f := \pi \circ \tilde{f}:S^{1} \to S^{1}$ fails to map a sufficiently small neighborhood of $z_{0}$ to an open set in $S^{1}$.

Particularly, $f$ cannot be a covering, or even an open map. Contrapositively, no covering map $f:S^{1} \to S^{1}$ lifts to a map $\tilde{f}:S^{1} \to \mathbf{R}$.

Answer 2

Fact(general lifting lemma) $p : (E, e_0) \to (B, b_0)$ be a covering map and $f : (X, x_0) \to (B, b_0)$ be a continuous map. $f$ lifts to $\widetilde{f} : (X, x_0) \to (E, e_0)$ (such that the obvious diagram commutes) if and only if $f_*(\pi_1(X, x_0)) \subset p_*(\pi_1(E, e_0))$.

That said, if $f$ is some finite-sheeted covering map $S^1 \to S^1$ and $p : \Bbb R \to S^1$ the universal covering of $S^1$, then $f_*(\pi_1(S^1))$ would be isomorphic to $\Bbb Z$ (as the only finite index subgroups of $\Bbb Z$ are $n\Bbb Z \cong \Bbb Z$), whereas $p_*(\pi_1(\Bbb R)) \cong 0$. Since $\Bbb Z$ doesn't inject onto the trivial group, there is no such lift.

As the identity map $\text{id} : S^1 \to S^1$ and the two-fold cover $S^1 \to S^1$ are both finite sheeted covering maps, that answer to the question is no.