Group and Lie algebra filtrations and homotopy groups of spheres

Abstract

AbstractWe 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 the torsion of the homotopy group into a dimension quotient, via a result of Wu. In particular, this invalidates some long‐standing results in the literature, as for every prime , there is some ‐torsion in by a result of Serre. We explain in this manner Rips's famous counterexample to the dimension conjecture in terms of the homotopy group . We finally obtain analogous results in the context of Lie rings: for every prime there exists a Lie ring with ‐torsion in some dimension quotient.