How can I show that there cannot exist a homotopy from $\mathcal{C_1}$ to $\mathcal{C_2}$? Consider the diagram below with an annulus $\mathcal{A}$ and two circles in the annulus $\mathcal{C}_1$ and $\mathcal{C}_2$. 
In $\mathbb{R^n}$, there clearly exists a homotopy between any two circles. But if we have this scenario here, where we are restricted by the annulus, it doesn't seem like there exists a homotopy from $\mathcal{C}_1$ to $\mathcal{C}_2$. This is just based on my intuition.
But how can I actually prove this?


They are homeomorphic. It doesn't really matter that they lie in the annulus, since the topology they inherit is just the same as the subspace of $\mathbb{R}^n$.
