Group and Lie algebra filtrations and homotopy groups of spheres

Abstract

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 𝜋𝑠(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 𝜋2𝑝(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 𝜋4(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.