Why is the fundamental group of the circle so non-trivial? I've recently learned that the fundamental group of the circle is isomorphic to $\mathbb{Z}$. I'm having a hard time picturing loops in $S^1$; can you give some intuition for why that group has such a rich structure?
 A: I think one thing that may be confusing you is that you are envisioning a "loop" in a space as a circle sitting inside the space.  This picture is incorrect: a loop is a map from a circle to your space, but this map does not need to be injective!  So your vision of a loop should be dynamic: it is a a path that you can trace in the space which may cross or repeat itself, not just a fixed subset of the space.
With this picture, the loops in $S^1$ corresponding to the elements of $\mathbb{Z}$ are quite simple.  Namely, a nonnegative integer $n$ corresponds to the path that goes all the way around the circle $n$ times counterclockwise.  Negative integers are similar, except you go around clockwise instead.  So even though a circle has only one "loop" to go around, a path can traverse that loop multiple times (and can go either forward or backwards), and that's how you end up with an entire $\mathbb{Z}$ in the fundamental group.
A: Imagine any loop $\gamma$  in the punctured plane $\dot{\mathbb R}^2$. The normalization map
$$\nu:\quad (x,y)\mapsto{1\over\sqrt{x^2+y^2}}(x,y)$$
converts $\gamma$ into the loop $\gamma':=\nu\circ\gamma$ on $S^1$. The given $\gamma$ has a winding number
$$N(\gamma,0)={1\over2\pi}\int_\gamma\nabla{\rm arg}({\bf z})\cdot d{\bf z}\qquad\left(={1\over 2\pi i}\int_\gamma{dz\over z}\right)$$
with respect to the origin, and this is also the winding number (or the degree) of $\gamma'\subset S^1$.
It is one of the most basic and all important facts of geometry and analysis that this degree characterizes $\gamma$, resp. $\gamma'$, up to homotopy – in other terms: that the fundamental group of $S^1$ is cyclic.
