The Freudenthal Suspension Theorem

The Freudenthal suspension theorem establishes that homotopy groups of spheres eventually stabilize, giving rise to the stable homotopy groups that form the foundation of stable homotopy theory.

The Suspension Map

Definition

The suspension of a pointed space $X$ is $\Sigma X = X \wedge S^1 = (X \times [0,1]) / (X \times \{0\} \cup X \times \{1\} \cup \{x_0\} \times [0,1])$ . The suspension homomorphism is $\Sigma : \pi_n(X) \to \pi_{n+1}(\Sigma X)$ defined by $\Sigma[f] = [\Sigma f]$ , where $\Sigma f : S^{n+1} \cong \Sigma S^n \to \Sigma X$ is obtained by suspending the map $f : S^n \to X$ .

Since $\Sigma S^n \cong S^{n+1}$ , the suspension gives maps $\Sigma : \pi_{n+k}(S^n) \to \pi_{n+k+1}(S^{n+1})$ between homotopy groups of spheres.

The Theorem

Theorem9.7Freudenthal Suspension Theorem

If $X$ is an $(n-1)$ -connected CW complex with $n \geq 1$ , then the suspension homomorphism $\Sigma : \pi_k(X) \to \pi_{k+1}(\Sigma X)$ is an isomorphism for $k < 2n - 1$ and a surjection for $k = 2n - 1$ .

Applying this to $X = S^n$ (which is $(n-1)$ -connected), we obtain:

Theorem9.8Stability of Homotopy Groups of Spheres

The suspension map $\Sigma : \pi_{n+k}(S^n) \to \pi_{n+k+1}(S^{n+1})$ is an isomorphism for $n > k + 1$ and a surjection for $n = k + 1$ . In particular, the sequence $\pi_{k+1}(S^1) \to \pi_{k+2}(S^2) \to \pi_{k+3}(S^3) \to \cdots$ stabilizes for $n$ sufficiently large (specifically, $n \geq k + 2$ ).

ExampleFirst stable homotopy group

For $k = 1$ : $\pi_3(S^2) \cong \mathbb{Z}$ , $\pi_4(S^3) \cong \mathbb{Z}/2$ , $\pi_5(S^4) \cong \mathbb{Z}/2$ , and $\pi_{n+1}(S^n) \cong \mathbb{Z}/2$ for all $n \geq 3$ . The stable value $\pi_1^s = \mathbb{Z}/2$ is generated by the suspension of the Hopf map $\eta : S^3 \to S^2$ .

Stable Homotopy Groups

Definition

The $k$ -th stable homotopy group of spheres is $\pi_k^s = \varinjlim_{n \to \infty} \pi_{n+k}(S^n) = \pi_{N+k}(S^N) \quad \text{for any } N \geq k + 2$ These groups form a graded ring under composition (and the smash product), called the stable homotopy ring $\pi_*^s$ .

RemarkComputational status

The stable homotopy groups of spheres are among the most studied and difficult objects in algebraic topology. They are known through a large (but finite) range of dimensions, computed using the Adams spectral sequence, the Adams-Novikov spectral sequence, and chromatic methods. The first few values are $\pi_0^s \cong \mathbb{Z}$ , $\pi_1^s \cong \mathbb{Z}/2$ , $\pi_2^s \cong \mathbb{Z}/2$ , $\pi_3^s \cong \mathbb{Z}/24$ .