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  "body": "\n## Statement\n\nLet $(X, x_0)$ be a pointed space, and let $n \\ge 2$.\nThen the [[homotopy group|P976WYEFMB]] $\\pi_n(X, x_0)$ is an [[abelian group|082GHNPJ53]].\n\n## Remarks\n\nRepresent two elements by maps $f, g \\colon I^n \\to X$ that send $\\partial I^n$ to $x_0$.\nUp to homotopy, $f$ can be shrunk to a small cube and $g$ to another small cube, with the map equal to $x_0$ elsewhere.\nBecause $n \\ge 2$, there is room in a second coordinate direction to slide the two cubes around each other, which turns $f + g$ into $g + f$.\nAlgebraically, the sums along the first and the second coordinate are two operations with a common unit that commute with each other.\nThe [[Eckmann–Hilton argument|EJJYCPWHPA]] shows that they agree and are commutative.\nThe 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.\n\n## Explanations\n\n### L1 human\n\nLoops in a space can be combined, and the order may matter, as in a figure eight.\nFor spheres of dimension two or more, there is always room to move one past the other, so the order never matters.\n\n### L2 human\n\nThe result explains why $\\pi_n$ for $n \\ge 2$ behaves much like homology, and why the [[Hurewicz theorem|ZA11N5XPZM]] can compare them directly.\nThe fundamental group still acts on $\\pi_n(X, x_0)$, so $\\pi_n$ is a module over the group ring $\\Z[\\pi_1(X, x_0)]$.\nThe same sliding argument shows that $\\pi_1$ of a topological group, or of any H-space, is abelian.\n",
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