Visualizing the fundamental group of a cylinder

Question

the cylinder has a fundamental group that is isomorphic to the additive group of the integers. The idea I had for visualizing this is that given a point $a$ on the cylinder 0 is the constant function from $[0,1]$ to $a$ and that a positive integer $n$ is a path that loops around the cylinder $n$ times in the same fashion that one wraps a wire around stick and goes back to $a$ while it's inverse is the same path but is going in the opposite direction. Is this a correct way of looking at it?

Answer 1

Yes, exactly. If you want to rigorously prove that $\pi_1$ of the cylinder is isomorphic to $\mathbb{Z}$, you would probably want to use that the cylinder deformation retracts to the circle... but intuitively yours is the right mental picture to have in mind. And with this mental picture of the fundamental group, it's also easy to see the group structure; for example, if you take a path wrapping $m$ times around the cylinder and concatenate it with a path wrapping $n$ times around the cylinder in the same direction, the result is a path wrapping $m + n$ times around the cylinder.