Determine all homotopy groups of spheres

Determine the homotopy groups π_k(S^n) of spheres S^n for all integers k ≥ n ≥ 2, providing a complete specification of π_k(S^n) across all dimensions.

Background

The chapter explains that while the fundamental group of Sn is trivial for n ≥ 2, higher homotopy groups capture richer structure. It highlights that π_2(S2) ≅ ℤ, but beyond this simple case, the landscape becomes complex: many π_k(Sn) are non-trivial for k ≥ n, and a comprehensive determination across all degrees remains unresolved.

This classical problem is central to algebraic topology, influencing deep areas such as stable homotopy theory and the study of topological defects in physics. A full characterization of π_k(Sn) would unify and complete a major thread of homotopy theory.

References

The surprise comes with the result that \pi_k(Sn) is non-trivial for most (but certainly not all) k \geq n \geq 2, and in fact mathematicians have not yet determined all the homotopy groups of spheres for arbitrary k and n.

— Algebraic Topology  (1304.7846 - Robins, 2013) in Subsection "Extensions and applications" within Section "Homotopy Theory"

When is $w_nk$ nonzero? For instance, when $p=2$ one may ask whether $w_nk \neq 0$ whenever $n$ is not of the form $2i-1$. More generally, for fixed $k$, are infinitely many of the classes $w_nk$ nontrivial? Does the number of vanishing classes $w_nk$ increase with $k$? For $p>2$ and $n>1$, do any of these classes vanish? For which values of $n$ does one obtain an infinite family? We leave these questions open for further study.

— An abelian model for the Goodwillie tower of the circle  (2609.31269 - Nervo, 25 Sep 2026) in Section 1, subsection “Early convergence”

What is the group structure of $\pi_*Sn_1$? In particular, what are the $2$-extensions that involve the first line of the spectral sequence above?

— An abelian model for the Goodwillie tower of the circle  (2609.31269 - Nervo, 25 Sep 2026) in Section 3, subsection “Metastable computations”