What's the meaning of this about the cyclic fundamental group?

Question

"You can also get a cyclic group of order p by attaching a disk to a circle by wrapping it around the circle p times (the fact that the fundamental group is Z/pZ follows from Van-Kampen’s theorem). " But I can't understand what's the figure looks like about" attaching a disk to a circle by wrapping it around the circle p times". Thank you.

Answer 1

Consider the map $f_p\colon S^1\to S^1$ given by $f(e^{i\theta})=e^{pi\theta}$. This 'wraps' the circle around itself $p$ times. Now glue the disk $D^2$ to $S^1$ on it's boundary via the map $f_p$, that is, glue the point $e^{i\theta}$ in $\partial D^2$ to the point $e^{pi\theta}$ in $S^1$.