A book reads:
Refer to the proof of the following assertion: Given a map $F\colon Y \times I \to S^1$ and a map $\tilde{F}\colon Y \times\{0\} \to \mathbb{R}$ lifting $F| Y\times\{0\}$, there is a unique map $\tilde{F}\colon Y \times I \to \mathbb{R}$ lifting $F$ and restricting to the given $\tilde{F}$ on $Y\times\{0\}$.
(a) Explain how the existence of a lift $F| Y\times\{0\}$ is used in the proof.
(b) Find an example of where the assertion fails if the existence of a lift of $F| Y\times\{0\}$ is not assumed.
What I am concerned about is the (b). Could anyone suggest me an example? I can't think of one myself.
The problem is uniqueness: there may be multiple lifts (and in fact there is always an infinite number of lifts if $Y \neq \emptyset$). For example if $Y$ is a point and $I \to S^1$ is a constant map, every point in the fiber over the image (something that looks like $\mathbb{Z}$ if the covering map is $t \mapsto e^{2\pi it}$) is a possible lift.