Group and Lie algebra filtrations and homotopy groups of spheres

TL;DR

This paper connects homotopy groups of spheres with group and Lie algebra filtrations, solving the dimension problem by embedding torsion of homotopy groups into dimension quotients, and revising previous assumptions in the field.

Contribution

It establishes a novel link between homotopy groups and commutator calculus, providing a converse to Sjogren's theorem and embedding torsion in dimension quotients.

Findings

Every abelian group of bounded exponent can be embedded in a dimension quotient.

Torsion in homotopy groups of spheres can be embedded into dimension quotients.

Counterexamples to the dimension conjecture are explained via homotopy groups.

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 $\pi_s(S^d)$ into a dimension quotient, via a result of Wu. In particular, this invalidates some long-standing results in the literature, since for every prime $p$, there is some $p$-torsion in $\pi_{2p}(S^2)$ by a result of Serre. We explain in this manner Rips's famous counterexample to the dimension conjecture in terms of the homotopy group $\pi_4(S^2)=\mathbb Z/2\mathbb Z$. 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.