Homotopy groups of $S^2$

At least at $p=2$, Toda's calculations do not go nearly that far up. There are later papers that go a little further, but his ``Composition methods in homotopy groups of spheres'' goes up to $n=19$. It is misleading to think of $S^2$ as particularly simple. There is an old theorem (Serre?) that if $X$ is any simply connected finite CW complex that is not contractible, such as $S^2$, then for each prime $p$ there are infinitely many $n$ such that $\pi_n(X)$ has $p$-torsion.

For a pointed space $X$ let $\Omega X$ denote the loop space of loops $S^1 \to X$ respecting base-points. For non-pathological $X$ we can equip $\Omega X$ with the compact-open topology and we have the natural identification $\pi_n(\Omega X) \cong \pi_{n+1}(X)$.

Now $\pi_1$ is pretty intuitive. It tells us how complicated loops in $X$ can be up to homotopy. Equivalently, it tells us about the connected components of $\Omega X$. But it doesn't tell us how complicated those connected components themselves are. To think about that we look at $\pi_1(\Omega X) \cong \pi_2(X)$. When this is nontrivial, it means that there are loops in $\Omega X$ (or "loops between loops") that are not homotopy-equivalent.

Similarly $\pi_3(X) \cong \pi_2(\Omega X) \cong \pi_1(\Omega^2 X)$ tells us about how complicated "loops between loops between loops" are. In other words, the fact that the higher homotopy groups of $S^2$ are nontrivial tells us that the iterated loop spaces of $S^2$ are complicated.

By contrast, every connected component of $\Omega S^1$ is contractible by looking at the universal cover.