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Let $A$ be a annulus $\{ (r,\theta) \vert 1\le r \le 2 , \theta \in [0,2\pi] \}$. Gluing $(1,0)$ and $(2,0)$ together. Then how to prove that $\{r=1 \}$ and $\{ r = 2 \}$ are not homotopic rel $(1,0) \thicksim (2,0)$.

The difficulty is, for example, let $h_t(s)$ is the homotopy such that $h_0$ is the outer circle and $h_1$ is the inner circle. Then with $t$ and $s$ very small $h_t(s)$ may go through the point $(1,0)$ to the other side of the annulus. Just like the diagram below.

enter image description here

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