Derived functors of non-additive functors and homotopy theory

TL;DR

This paper introduces a functorial algebraic approach to understanding homotopy groups of spheres and Moore spaces, utilizing spectral sequences and Lie functor decompositions to compute torsion components and homotopy groups.

Contribution

It develops a new functorial framework for analyzing homotopy groups using algebraic tools like spectral sequences and Lie functor decompositions, providing explicit calculations.

Findings

Retrieved the 3-torsion component of π_i(S^2) up to degree 14

Unified presentation of homotopy groups π_i(M(A,n)) for small n and i

Provided a functorial algebraic method for homotopy group computations

Abstract

We develop a functorial approach to the study of the homotopy groups of spheres and Moore spaces $M(A,n)$, based on the Curtis spectral sequence and the decomposition of Lie functors as iterates of simpler functors such as the symmetric or exterior algebra functors. The discussion takes place over the integers, and includes a functorial description of the derived functors of certain Lie algebra functors, as well as of all the main cubical functors (such as the degree 3 component $SP^3$ of the symmetric algebra functor). As an illustration of this method, we retrieve in a purely algebraic manner the 3-torsion component of the homotopy groups of the 2-sphere up to degree 14, and give a unified presentation of homotopy groups $\pi_i(M(A,n))$ for small values of both $n$ and $i$.