TITLE: What is the intuition for higher homotopy groups not vanishing?
QUESTION [33 upvotes]: The homotopy groups of the spheres $S^n$ (see Wikipedia) vanish for the circle $S^1$ as, naively speaking, there are not higher order holes to be grasped by higher order homotopy groups. This intuitions already breaks down for the two sphere $S^2$, e.g. $\pi_3(S^2)$ is non-trivial because of the Hopf fibration. This non-triviality seems to keep on going for all the higher spheres $S^n$. What makes $S^1$ so fundamentally different?

REPLY [9 votes]: Here's a very simple argument as to why not all of the spheres have higher homotopy groups vanish. Higher homotopy groups vanishing (along with the lower ones) is equivalent to the sphere representing n-dimensional cohomology. By Yoneda, natural transformations between functors correspond to maps between representing objects. Hence, natural transformations between nth cohomology with integral coefficients and mth cohomology with integral coefficients correspond to maps $S^n \rightarrow S^m$ if these both represent singular cohomology. However, there are nontrivial cohomology operations (say for example, reduction mod p followed by the integral Bockstein), and so this implies that we have a map between the spheres representing them. However, this would imply that the higher homotopy groups of this sphere do not vanish, and so we get a contradiction.