Function doesn't have a lift in a space related to Topologist's sine curve

Question

I'm trying to solve exercise 1.3.7 in Hatcher's Algebraic Topology:

Let $Y$ be the quasi-circle that is the union of a portion of the graph $y = \sin(1/x)$, the line segment $[-1,1]$ in the $y$-axis, and an arc connecting the two pieces. Collapsing the segment in the $y$-axis to a point gives a quotient map $f:Y \to S^1$. Show that $f$ does not lift to the covering space $p:\mathbb{R} \to S^1$, even though $\pi_1(Y) = 0$.

I was able to show that $\pi_1(Y)$ is simply-connected by decomposing it into 2 pieces and showing that a path in $Y$ must be contained in a subspace that is homeomorphic to $\mathbb{R}$. What I'm having difficulty with is showing that $f$ doesn't have a lift. I've shown that if it has a lift $\tilde f$, $\tilde{f}([-1,1])$ is one point. I feel that this contradicts with the continuity of $\tilde f$, but I cannot rigorously prove it.

Any help would be appreciated. This is self-study by the way.

Answer 1

I'll use the convention that $S^1 = \{z\in\mathbb C : \lvert z \rvert = 1\}$ and $p(t)=e^{i t}$, to avoid the clash of notations between ordered pairs in $\mathbb R^2$ and open intervals in $\mathbb R$.

Let $L$ be the segment on the $y$-axis. WLOG assume that $f(L)=\{1\}$. Suppose $\tilde f\colon Y\rightarrow \mathbb R$ is a lift. Since $Y\setminus L$ is connected, $\tilde f(Y\setminus L)$ is also connected, so it must lie in one of the components of $p^{-1}(f(Y\setminus L))=\mathbb R \setminus 2\pi \mathbb Z$, say the interval $(0,2\pi)$. It follows by the surjectivity of $f$ that $\tilde f(Y\setminus L)$ must be precisely $(0,2\pi)$. Now $Y$ is compact, so $\tilde f(Y) \supset [0,2\pi]$. Thus, $\{0,2\pi\}\subset \tilde f(L)$, contradicting your observation that $\tilde f(L)$ is one point.

Alternatively for the last step, observe that $\tilde f(L)$ should be connected, but $\tilde f(L) \subset 2\pi\mathbb Z$ is a discrete set containing at least two points.

Answer 2

Consider the injective path $g:[0,\infty)\to Y$ starting at $(0,0)$, running through the arc connecting $(0,0)$ and $(1/\pi,0)$, and sending $x$ to $(1/x,\sin x)$ for $x\ge\pi$. Assume that $\tilde f:Y\to R$ is a lift for $f:Y\twoheadrightarrow S^1$. As $f$ is injective except for the vertical segment, and $g$ avoids this segment, $\tilde fg$ must be injective, and thus strictly increasing. Denote the point $\tilde fg(0)=\tilde f(0,0)$ by $y_0$. Consider the points $x_n=\left(\frac1{n\pi},0\right)$. We have $x_n=g(n\pi)$, so the sequence $y_n=\tilde f(x_n)$ increases. But $(x_n)_n\to (0,0)$, so $y_n\to y_0$, which is impossible as $y_0=\tilde fg(0)$ is less than $y_1=\tilde fg(\pi)$.