Stable homotopy groups of spheres

Summary

The stable homotopy groups of spheres $\pi_k^s$ are the groups $\pi_{n+k}(S^n)$ for $n$ large, where they no longer depend on $n$. The Freudenthal suspension theorem shows that the suspension maps become isomorphisms once $n \ge k + 2$. Together they form a graded ring, the coefficient ring of stable homotopy theory, and they are the homotopy groups of the sphere spectrum. Their computation is a central problem of algebraic topology, studied with the Adams spectral sequence and chromatic methods.