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(Sd) 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(S2) 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(S2)=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.