---
tag: M676G2V5KQ
kind: statement
role: proposition
name: Higher homotopy groups are abelian
msc: [55Q05]
uses: [P976WYEFMB, 082GHNPJ53]
about: [P976WYEFMB]
sources:
  - url: https://en.wikipedia.org/wiki/Homotopy_group
---

## Statement

Let $(X, x_0)$ be a pointed space, and let $n \ge 2$.
Then the [[homotopy group|P976WYEFMB]] $\pi_n(X, x_0)$ is an [[abelian group|082GHNPJ53]].

## 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|EJJYCPWHPA]] 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.

## Explanations

### L1 human

Loops in a space can be combined, and the order may matter, as in a figure eight.
For spheres of dimension two or more, there is always room to move one past the other, so the order never matters.

### L2 human

The result explains why $\pi_n$ for $n \ge 2$ behaves much like homology, and why the [[Hurewicz theorem|ZA11N5XPZM]] 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.