Let $D_1,D_2$ be 2-dimensional discs, whose boundaries are $S_1$ and $S_2$. Paste $e^{2\pi it}$ in $S_1$ to $e^{2n\pi it}$ in $S_2$, and then we get a quotient space $X$. Prove that $X$ is simply connected.
If $n=1$, then $X=S_2$, so it's easy to imagine $X$ is path connected. But when $n\geqslant2$, the topology in X becomes complex, and I don't know how to construct a homotopy (because it needs to be continuous) between a loop and a constant loop.
Or should I use the fact that $S_2$ is simply connected?
For instance, since the subspace $D_1\subseteq X$ is contractible we have a homotopy equivalence $X\simeq X/D_1$. But $X/D_1 \equiv S^2$.