As I have been studying algebraic topology, something that I found puzzling was the existence of finite homotopy groups. For instance, $\pi\_{4}(S^{2})\cong\pi\_{5}(S^{4})\cong\mathbb{Z}/2\mathbb{Z}$. I was wondering if there was any kind of intuitive reason for why this might be true, and if there are spaces $X$ such that $\pi\_{1}(X)$ is finite. Speaking very roughly, it would seem that a finite, nontrivial fundamental group means that if you repeat a closed path enough times, it can be contracted to a point, something which I find rather hard to visualize. So the question is: Is there any intuitive reason for the existence of finite homotopy groups?



Intuition on finite homotopy groups
The simplest (to understand) case of finite $\pi\_1$ is the group $SO\_3$. This can be illustrated using an arm or a belt! $SO\_3$ is the group of rotations in space and a based loop in $SO\_3$ can be thought of as a description of the motion of an object in such a way that it ends up back where it started. By attaching a strip of paper to the object, it's possible to see this path in space. For example, taking a belt and holding one end fixed whilst moving the other, or moving your hand (your arm forms the "strip").


So: hold your hand out in front of you, this is easiest if you do it palm-up. Keeping it palm-up, rotate it *under* your arm back to where you started. Your arm is now twisted (hopefully not too badly). Continue moving your hand **in the same direction** and with your palm facing up but this time over the top of your arm. When your hand gets back to where it started, your arm is now magically untwisted! So **two** times round the loop gets you back to an untwisted state, thus $2\gamma = 0$. But $\gamma \ne 0$ as evidenced by your twisted arm at the half-way stage.


If you find this difficult to do, here's an alternative way using a belt. Take a belt and twist it once (that is, hold it out straight and imagine an axis along its length, then twist one end all the way around). Now try to straighten it *without* twisting either end (though you can move either end in space). Can't be done. But if you twist the belt *twice* then it can.


(There's some funky you-tube videos showing the arm twists. If you get really good at it, you should do it with a beaker of water.)


