Intuition behind the fundamental group $\pi_1(S^1)$

Question

I have gone over the fact that $\pi_1(S^1) \cong \mathbb{Z}$ and its proof, though I admittedly do not understand it in full. What I am struggling with is the intuition behind this fact, since as I understand it, every integer $n$ can be thought of as how many revolutions we make around the circle. However in view of the intuition behind homotopy and deformations, regardless of how many times we loop through the circle, is the deformation not going to be the same, that is, result in $S^1$? How then do different $n$ represent distinct homotopy classes?

Answer 1

I think the key to your misunderstanding is that a "loop" in $S^1$ in the definition of the fundamental group is a function $l:[0,1]\to S^1$, not a subset of $S^1$. This is quite a subtle point, but absolutely crucial. If a loop was defined as a sunset (in particular as the image of the function $l$, what is normally referred to as the trace of $l$) then you would be right, a loop which goes around $S^1$ once is exactly the same as a loop which goes around it a million times. However, a function which wraps $[0,1]$ around $S^1$ is completely different to a function which wraps $[0,1]$ around $S^1$ a million times.

Let's be concrete to illustrate this. Identify $S^1$ with the unit circle in $\mathbb{C}$ and define $l_1:[0,1]\to S^1:t\mapsto e^{2\pi i t}$, and define $l_2:[0,1]\to S^1:t\mapsto e^{2\pi i (2t)}$. These both trace out $S^1$, so their images are equal as sets. However, at $t=1/2$ for example, $l_1(1/2)=-1$, while $l_2(1/2)=1$, so these are different as functions. In particular we can think of $l_1$ as representing $1$ in $\pi_1(S^1)$, and $l_2$ as representing $2$.