Stable homotopy groups of spheres

Remarks

By the Freudenthal suspension theorem, $E \colon \pi_{n+k}(S^n) \to \pi_{n+k+1}(S^{n+1})$ is an isomorphism for $n \ge k + 2$. So $\pi_k^s \cong \pi_{n+k}(S^n)$ for every $n \ge k + 2$.

Example: $\pi_0^s \cong \Z$, because $\pi_n(S^n) \cong \Z$ for all $n$ and $E$ preserves the degree.

Example: $\pi_1^s \cong \Z/2$, generated by the suspensions of the Hopf map $S^3 \to S^2$; the group $\pi_3(S^2) \cong \Z$ itself is not yet in the stable range, since $n = 2 < k + 2 = 3$.

The next values are $\pi_2^s \cong \Z/2$, $\pi_3^s \cong \Z/24$, $\pi_4^s = \pi_5^s = 0$, $\pi_6^s \cong \Z/2$ and $\pi_7^s \cong \Z/240$. Composition of maps makes $\pi_*^s = \bigoplus_k \pi_k^s$ a graded commutative ring. For $k \ge 1$, the group $\pi_k^s$ is finite.