Higher Homotopy Groups

The higher homotopy groups $\pi_n(X)$ for $n \geq 2$ generalize the fundamental group by using maps from higher-dimensional spheres instead of loops. Unlike $\pi_1$ , these groups are always abelian.

Definition

Let $X$ be a pointed topological space with basepoint $x_0$ . The $n$ -th homotopy group of $X$ is $\pi_n(X, x_0) = [(S^n, s_0), (X, x_0)]$ the set of basepoint-preserving homotopy classes of maps $f : (S^n, s_0) \to (X, x_0)$ . Equivalently, viewing $S^n = I^n / \partial I^n$ , $\pi_n(X, x_0) = \{f : I^n \to X \mid f(\partial I^n) = x_0\} / \simeq$ where $\simeq$ denotes homotopy relative to $\partial I^n$ .

The group operation is defined by "concatenation in the first coordinate": $([f] + [g])(t_1, t_2, \ldots, t_n) = \begin{cases} f(2t_1, t_2, \ldots, t_n) & 0 \leq t_1 \leq 1/2 \\ g(2t_1 - 1, t_2, \ldots, t_n) & 1/2 \leq t_1 \leq 1 \end{cases}$

Definition

A space $X$ is $n$ -connected if $\pi_k(X) = 0$ for all $k \leq n$ . A space is simply connected if $\pi_0(X) = 0$ (path-connected) and $\pi_1(X) = 0$ . A weakly contractible space has $\pi_n(X) = 0$ for all $n \geq 0$ .

Commutativity

Theorem9.1Abelianness of Higher Homotopy Groups

For $n \geq 2$ , the group $\pi_n(X, x_0)$ is abelian. That is, for any two maps $f, g : S^n \to X$ , $[f] + [g] = [g] + [f]$

The proof uses the Eckmann-Hilton argument: for $n \geq 2$ , there are two independent directions in which to concatenate, and interchanging them provides a homotopy between $f + g$ and $g + f$ .

ExampleHomotopy groups of spheres

The computation of $\pi_n(S^m)$ is one of the central problems in algebraic topology. Key known values include:

$\pi_n(S^n) \cong \mathbb{Z}$ for all $n \geq 1$ (generated by the identity map)
$\pi_3(S^2) \cong \mathbb{Z}$ (generated by the Hopf fibration)
$\pi_{n+1}(S^n) \cong \mathbb{Z}/2\mathbb{Z}$ for $n \geq 3$
$\pi_{n+2}(S^n) \cong \mathbb{Z}/2\mathbb{Z}$ for $n \geq 2$

The full computation remains an open problem; the structure is far more complex than homology.

RemarkDifficulty of computation

Unlike homology, homotopy groups are extremely difficult to compute. There is no general algorithm for computing $\pi_n(X)$ , and the homotopy groups of even the simplest spaces (like spheres) exhibit intricate and unpredictable patterns. This difficulty reflects the fundamentally non-linear nature of homotopy theory.