Higher homotopy groups are abelian

Remarks

Represent two elements by maps $f, g \colon I^n \to X$ that send $\partial I^n$ to $x_0$. Up to homotopy, $f$ can be shrunk to a small cube and $g$ to another small cube, with the map equal to $x_0$ elsewhere. Because $n \ge 2$, there is room in a second coordinate direction to slide the two cubes around each other, which turns $f + g$ into $g + f$. Algebraically, the sums along the first and the second coordinate are two operations with a common unit that commute with each other. The Eckmann–Hilton argument shows that they agree and are commutative. The hypothesis $n \ge 2$ is necessary: the fundamental group of the wedge of two circles is the free group on two generators, which is not abelian.

Going deeper

For a student

The result explains why $\pi_n$ for $n \ge 2$ behaves much like homology, and why the Hurewicz theorem can compare them directly. The fundamental group still acts on $\pi_n(X, x_0)$, so $\pi_n$ is a module over the group ring $\Z[\pi_1(X, x_0)]$. The same sliding argument shows that $\pi_1$ of a topological group, or of any H-space, is abelian.