Is there any 'nice' space with fundamental group $\mathbb{Z}_3$?

Question

I'm trying to build up intuition for the fundamental group, as it occurs in physics.

In the simplest examples, the fundamental group is trivial, or $\mathbb{Z}$, or $\mathbb{Z}^n$. We can also get $\mathbb{Z}_2$, for example in $SO(3)$, when traversing a path twice makes it homotopic to the identity.

I think the latter example is interesting, but I'm having trouble generalizing it so that traversing a path three or four times gives the identity. Are there "nice" spaces with fundamental group $\mathbb{Z}_n$, for any $n$? Or can you only get $\mathbb{Z}_2$ and products thereof?

Since I'm a physics major, I prefer concrete examples if possible, ideally a space one can faithfully draw on a piece of paper.

Answer 1

The $k$-fold dunce cap (the generalization of the $3$-fold dunce cap as seen in this question) is probably the simplest example. It's easy to draw faithfully, as long as you're comfortable with the idea of identifying edges:

(Be warned: there are a few different spaces often called the "dunce cap"; only one of them has fundamental group $\Bbb{Z}/k$.)

Answer 2

You are looking for lens spaces.

One can draw them:

(The image comes from https://plus.maths.org/content/dont-judge-black-hole-its-area-2, and there you can find an explanation of what it means)