Group and Lie algebra filtrations and homotopy groups of spheres

We establish a bridge between homotopy groups of spheres and commutator calculus in groups, and solve in this manner the “dimension problem” by providing a converse to Sjogren's theorem: every abelian group of bounded exponent can be embedded in the dimension quotient of a group. This is proven by embedding for arbitrary $s,d$ the torsion of the homotopy group ${\pi }_s\left(S^d\right)$ into a dimension quotient, via a result of Wu. In particular, this invalidates some long-standing results in the literature, as for every prime $p$, there is some $p$-torsion in ${\pi }_{2p}\left(S^2\right)$ by a result of Serre. We explain in this manner Rips's famous counterexample to the dimension conjecture in terms of the homotopy group ${\pi }_4\left(S^2\right)=Z/2Z$. We finally obtain analogous results in the context of Lie rings: for every prime $p$ there exists a Lie ring with $p$-torsion in some dimension quotient.